9 42

9_41

9_43

Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42's page at Knotilus! Visit 9 42's page at the original Knot Atlas! 9 42 Quick Notes

9_42 is Alexander Stoimenow's favourite knot!

Alsacian chair, alsacian museum, Strasbourg, France

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -18 -6 -12
Conway Notation	[22,3,2-]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-5] Hyperbolic Volume 4.05686 A-Polynomial See Data:9 42/A-polynomial

[edit Notes for 9 42's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant 0

[edit Notes for 9 42's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+2 t-1+2 t^{-1} - t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4-2 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 7, 2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-q^2+q-1+ q^{-1} - q^{-2} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6-5 a z^5-5 z^5 a^{-1} -5 a^2 z^4-5 z^4 a^{-2} -10 z^4+6 a z^3+6 z^3 a^{-1} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}+q^8+q^6-q^2-1- q^{-2} + q^{-6} + q^{-8} + q^{-10} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{46}+q^{42}+2 q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^8-q^4-2 q^2-1- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-22} + q^{-24} + q^{-26} +3 q^{-30} + q^{-34} + q^{-36} + q^{-40} + q^{-46} - q^{-50} - q^{-54} + q^{-56} - q^{-60} + q^{-62} }

Further Quantum Invariants

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^7+ q^{-7} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{18}+q^6+1+ q^{-6} - q^{-18} + q^{-22} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{45}-q^{41}-q^{39}+q^{35}+q^{23}+q^{21}-q^{19}-q^{17}+q^{13}-q^9+q^5+q^3+ q^{-3} + q^{-5} - q^{-13} - q^{-15} + q^{-21} +2 q^{-23} - q^{-27} +2 q^{-31} -3 q^{-35} - q^{-37} + q^{-39} + q^{-41} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{76}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}-q^{58}+q^{48}+q^{46}-2 q^{42}-2 q^{40}+q^{36}+2 q^{34}-q^{30}-q^{28}+2 q^{24}+q^{22}+q^{20}-q^{18}-2 q^{16}+q^{12}+2 q^{10}-2 q^6-q^4+2+ q^{-2} - q^{-4} + q^{-8} +2 q^{-10} + q^{-12} - q^{-14} + q^{-20} - q^{-24} - q^{-34} -2 q^{-36} + q^{-40} +2 q^{-42} +2 q^{-44} - q^{-46} - q^{-48} -2 q^{-50} + q^{-52} +3 q^{-54} -2 q^{-60} - q^{-62} + q^{-68} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{115}-q^{111}-q^{109}-q^{107}+q^{103}+2 q^{101}+q^{99}-q^{95}-q^{93}-q^{91}+q^{87}+q^{81}+q^{79}-q^{75}-3 q^{73}-2 q^{71}+2 q^{67}+3 q^{65}+2 q^{63}-2 q^{59}-2 q^{57}-q^{55}+q^{53}+2 q^{51}+2 q^{49}+q^{47}-q^{45}-2 q^{43}-3 q^{41}-2 q^{39}+q^{37}+3 q^{35}+3 q^{33}+q^{31}-2 q^{29}-4 q^{27}-2 q^{25}+q^{23}+3 q^{21}+4 q^{19}+q^{17}-2 q^{15}-3 q^{13}-q^{11}+2 q^9+3 q^7+2 q^5-q^3-3 q-2 q^{-1} + q^{-3} +3 q^{-5} +2 q^{-7} -2 q^{-11} - q^{-13} + q^{-15} +2 q^{-17} + q^{-19} - q^{-21} - q^{-23} + q^{-27} - q^{-31} - q^{-33} + q^{-37} + q^{-39} + q^{-55} +2 q^{-57} + q^{-59} -2 q^{-61} -4 q^{-63} -4 q^{-65} +5 q^{-69} +6 q^{-71} + q^{-73} -5 q^{-75} -6 q^{-77} -3 q^{-79} +3 q^{-81} +6 q^{-83} +5 q^{-85} -3 q^{-89} -3 q^{-91} -2 q^{-93} + q^{-97} + q^{-99} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}+q^8+q^6-q^2-1- q^{-2} + q^{-6} + q^{-8} + q^{-10} }
1,1	$q^{28}+2q^{24}+2q^{20}-2q^{18}-2q^{16}-2q^{14}-2q^{12}+2q^{6}+4q^{4}+4q^{2}+2+2q^{-2}-2q^{-4}-4q^{-8}-2q^{-12}+2q^{-14}+2q^{-18}+2q^{-24}-2q^{-26}+q^{-28}-2q^{-30}+2q^{-32}$
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{26}+q^{24}-q^{18}-q^{16}-2 q^{14}-2 q^{12}-q^{10}+q^8+3 q^6+2 q^4+3 q^2+2+ q^{-2} - q^{-4} - q^{-6} - q^{-8} - q^{-10} + q^{-16} + q^{-20} - q^{-22} - q^{-24} + q^{-28} + q^{-30} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}+q^{16}+q^{14}+q^{12}-q^4- q^{-4} + q^{-12} + q^{-14} + q^{-16} + q^{-20} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{13}+q^{11}+2 q^9+q^7-q^3-2 q-2 q^{-1} - q^{-3} + q^{-7} +2 q^{-9} + q^{-11} + q^{-13} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}+q^{24}+2 q^{22}+2 q^{20}+2 q^{18}+q^{16}-2 q^{12}-3 q^{10}-3 q^8-2 q^6-q^4+q^2+4+4 q^{-2} +3 q^{-4} +2 q^{-6} -3 q^{-10} -3 q^{-12} -3 q^{-14} -2 q^{-16} +2 q^{-20} +3 q^{-22} +2 q^{-24} +2 q^{-26} + q^{-28} - q^{-32} }
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^8-q^4-2 q^2-3-2 q^{-2} - q^{-4} + q^{-8} +2 q^{-10} +2 q^{-12} + q^{-14} + q^{-16} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}+q^{16}+q^{14}+q^{12}-q^4-2- q^{-4} + q^{-12} + q^{-14} + q^{-16} + q^{-20} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{34}+q^{26}+q^{18}-q^{14}+1- q^{-14} + q^{-18} + q^{-26} + q^{-34} }

D4 Invariants.

Weight	Invariant
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14}+q^{12}-q^6-q^4-2 q^2-2 q^{-2} - q^{-4} - q^{-6} + q^{-12} + q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-22} + q^{-26} }

G2 Invariants.

Weight	Invariant
1,0	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 42"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+2 t-1+2 t^{-1} - t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4-2 z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 7, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-q^2+q-1+ q^{-1} - q^{-2} + q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4+a^2 z^2+z^2 a^{-2} -4 z^2+2 a^2+2 a^{-2} -3}

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6-5 a z^5-5 z^5 a^{-1} -5 a^2 z^4-5 z^4 a^{-2} -10 z^4+6 a z^3+6 z^3 a^{-1} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-2 a z-2 z a^{-1} -2 a^2-2 a^{-2} -3}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}

Vassiliev invariants

V2 and V3:

(-2, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} ${\frac {164}{3}}$ ${\frac {76}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1312}{3}}$ $-{\frac {608}{3}}$ $-{\frac {6271}{15}}$ ${\frac {1484}{15}}$ $-{\frac {19564}{45}}$ ${\frac {607}{9}}$ $-{\frac {1471}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 9 42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 χ 7             1 1 5               0 3         1 1   0 1       1 1     0 -1       1 1     0 -3   1 1         0 -5               0 -7 1             1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-q^9-q^8+2 q^7-q^6-q^5+2 q^4-q^3+q-1+ q^{-1} - q^{-3} +2 q^{-4} - q^{-5} - q^{-6} +2 q^{-7} - q^{-8} - q^{-9} + q^{-10} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{19}-2 q^{17}-2 q^{16}+4 q^{15}+2 q^{14}-4 q^{13}-3 q^{12}+5 q^{11}+4 q^{10}-5 q^9-4 q^8+5 q^7+3 q^6-5 q^5-3 q^4+5 q^3+3 q^2-4 q-3+4 q^{-1} +3 q^{-2} -3 q^{-3} -3 q^{-4} +3 q^{-5} +2 q^{-6} -2 q^{-7} -2 q^{-8} +2 q^{-9} + q^{-10} -2 q^{-11} +2 q^{-13} -2 q^{-15} +2 q^{-17} - q^{-19} - q^{-20} + q^{-21} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-q^{31}-q^{29}-q^{28}+3 q^{27}-q^{26}+3 q^{25}-3 q^{24}-4 q^{23}+4 q^{22}-q^{21}+6 q^{20}-3 q^{19}-5 q^{18}+3 q^{17}-3 q^{16}+7 q^{15}-2 q^{14}-5 q^{13}+3 q^{12}-3 q^{11}+6 q^{10}-q^9-4 q^8+2 q^7-3 q^6+5 q^5+q^4-3 q^3+q^2-4 q+4+3 q^{-1} -2 q^{-2} - q^{-3} -5 q^{-4} +3 q^{-5} +5 q^{-6} -2 q^{-8} -6 q^{-9} + q^{-10} +6 q^{-11} +2 q^{-12} -2 q^{-13} -5 q^{-14} - q^{-15} +5 q^{-16} +2 q^{-17} - q^{-18} -3 q^{-19} -2 q^{-20} +4 q^{-21} - q^{-23} - q^{-24} - q^{-25} +4 q^{-26} - q^{-27} - q^{-28} - q^{-29} - q^{-30} +3 q^{-31} - q^{-34} - q^{-35} + q^{-36} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{47}-q^{45}-2 q^{44}-q^{43}+4 q^{41}+5 q^{40}-5 q^{38}-7 q^{37}-3 q^{36}+5 q^{35}+11 q^{34}+5 q^{33}-6 q^{32}-12 q^{31}-7 q^{30}+5 q^{29}+13 q^{28}+8 q^{27}-5 q^{26}-13 q^{25}-8 q^{24}+5 q^{23}+13 q^{22}+8 q^{21}-5 q^{20}-13 q^{19}-8 q^{18}+6 q^{17}+13 q^{16}+7 q^{15}-6 q^{14}-13 q^{13}-7 q^{12}+7 q^{11}+12 q^{10}+6 q^9-6 q^8-11 q^7-6 q^6+6 q^5+10 q^4+5 q^3-4 q^2-9 q-5+4 q^{-1} +7 q^{-2} +4 q^{-3} -2 q^{-4} -6 q^{-5} -4 q^{-6} +3 q^{-7} +4 q^{-8} +2 q^{-9} - q^{-10} -3 q^{-11} - q^{-12} +2 q^{-13} +2 q^{-14} - q^{-15} -3 q^{-16} - q^{-17} +2 q^{-18} +4 q^{-19} +2 q^{-20} -3 q^{-21} -6 q^{-22} -2 q^{-23} +3 q^{-24} +5 q^{-25} +4 q^{-26} -2 q^{-27} -6 q^{-28} -3 q^{-29} + q^{-30} +4 q^{-31} +4 q^{-32} -4 q^{-34} -2 q^{-35} +2 q^{-37} +2 q^{-38} - q^{-39} -2 q^{-40} - q^{-41} + q^{-42} +2 q^{-43} + q^{-44} - q^{-45} - q^{-46} -2 q^{-47} +2 q^{-49} + q^{-50} - q^{-53} - q^{-54} + q^{-55} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-q^{65}-q^{62}-q^{61}-q^{60}+3 q^{59}+q^{58}+4 q^{57}+q^{56}-2 q^{55}-7 q^{54}-6 q^{53}+2 q^{51}+14 q^{50}+7 q^{49}+2 q^{48}-13 q^{47}-14 q^{46}-7 q^{45}-q^{44}+20 q^{43}+13 q^{42}+9 q^{41}-12 q^{40}-16 q^{39}-13 q^{38}-5 q^{37}+20 q^{36}+14 q^{35}+13 q^{34}-11 q^{33}-14 q^{32}-15 q^{31}-7 q^{30}+19 q^{29}+13 q^{28}+14 q^{27}-11 q^{26}-14 q^{25}-15 q^{24}-6 q^{23}+19 q^{22}+13 q^{21}+14 q^{20}-10 q^{19}-14 q^{18}-15 q^{17}-5 q^{16}+17 q^{15}+11 q^{14}+14 q^{13}-8 q^{12}-12 q^{11}-15 q^{10}-5 q^9+13 q^8+9 q^7+16 q^6-5 q^5-9 q^4-16 q^3-6 q^2+8 q+7+18 q^{-1} -5 q^{-3} -17 q^{-4} -8 q^{-5} +2 q^{-6} +4 q^{-7} +19 q^{-8} +5 q^{-9} -16 q^{-11} -8 q^{-12} -4 q^{-13} - q^{-14} +17 q^{-15} +8 q^{-16} +4 q^{-17} -12 q^{-18} -5 q^{-19} -7 q^{-20} -5 q^{-21} +12 q^{-22} +7 q^{-23} +5 q^{-24} -8 q^{-25} -5 q^{-27} -5 q^{-28} +8 q^{-29} +3 q^{-30} + q^{-31} -7 q^{-32} +2 q^{-33} - q^{-34} - q^{-35} +8 q^{-36} +2 q^{-37} -3 q^{-38} -8 q^{-39} - q^{-40} - q^{-41} + q^{-42} +9 q^{-43} +4 q^{-44} -2 q^{-45} -6 q^{-46} -3 q^{-47} -3 q^{-48} - q^{-49} +8 q^{-50} +3 q^{-51} -2 q^{-53} -2 q^{-54} -2 q^{-55} -2 q^{-56} +6 q^{-57} - q^{-59} - q^{-60} - q^{-61} - q^{-62} - q^{-63} +5 q^{-64} - q^{-67} - q^{-68} -2 q^{-69} - q^{-70} +3 q^{-71} + q^{-73} - q^{-76} - q^{-77} + q^{-78} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{87}-q^{85}-q^{84}-q^{83}-q^{82}+q^{80}+5 q^{79}+4 q^{78}+q^{77}-q^{76}-5 q^{75}-7 q^{74}-9 q^{73}-4 q^{72}+7 q^{71}+15 q^{70}+11 q^{69}+9 q^{68}-q^{67}-15 q^{66}-22 q^{65}-20 q^{64}+q^{63}+18 q^{62}+23 q^{61}+24 q^{60}+8 q^{59}-15 q^{58}-27 q^{57}-30 q^{56}-10 q^{55}+15 q^{54}+25 q^{53}+30 q^{52}+13 q^{51}-12 q^{50}-24 q^{49}-31 q^{48}-13 q^{47}+12 q^{46}+23 q^{45}+29 q^{44}+13 q^{43}-13 q^{42}-22 q^{41}-29 q^{40}-12 q^{39}+13 q^{38}+22 q^{37}+29 q^{36}+12 q^{35}-13 q^{34}-23 q^{33}-29 q^{32}-11 q^{31}+14 q^{30}+23 q^{29}+28 q^{28}+11 q^{27}-14 q^{26}-24 q^{25}-27 q^{24}-9 q^{23}+15 q^{22}+22 q^{21}+25 q^{20}+9 q^{19}-14 q^{18}-22 q^{17}-24 q^{16}-7 q^{15}+13 q^{14}+19 q^{13}+21 q^{12}+8 q^{11}-11 q^{10}-17 q^9-19 q^8-7 q^7+9 q^6+13 q^5+16 q^4+9 q^3-6 q^2-10 q-14-7 q^{-1} +3 q^{-2} +5 q^{-3} +10 q^{-4} +9 q^{-5} -3 q^{-7} -6 q^{-8} -6 q^{-9} -2 q^{-10} -3 q^{-11} + q^{-12} +6 q^{-13} +3 q^{-14} +5 q^{-15} +3 q^{-16} - q^{-17} -2 q^{-18} -8 q^{-19} -9 q^{-20} -2 q^{-21} + q^{-22} +8 q^{-23} +11 q^{-24} +7 q^{-25} +4 q^{-26} -7 q^{-27} -14 q^{-28} -10 q^{-29} -6 q^{-30} +3 q^{-31} +12 q^{-32} +13 q^{-33} +10 q^{-34} -11 q^{-36} -11 q^{-37} -11 q^{-38} -4 q^{-39} +7 q^{-40} +10 q^{-41} +10 q^{-42} +4 q^{-43} -4 q^{-44} -6 q^{-45} -7 q^{-46} -5 q^{-47} +3 q^{-48} +5 q^{-49} +4 q^{-50} + q^{-51} -3 q^{-52} -3 q^{-53} -2 q^{-54} +4 q^{-56} +6 q^{-57} +2 q^{-58} -2 q^{-59} -6 q^{-60} -6 q^{-61} -2 q^{-62} +4 q^{-64} +8 q^{-65} +5 q^{-66} -4 q^{-68} -7 q^{-69} -3 q^{-70} -2 q^{-71} +6 q^{-73} +4 q^{-74} +2 q^{-75} - q^{-76} -4 q^{-77} - q^{-78} - q^{-79} - q^{-80} +4 q^{-81} + q^{-82} -3 q^{-85} - q^{-86} - q^{-87} +4 q^{-89} + q^{-90} + q^{-92} -2 q^{-93} - q^{-94} -2 q^{-95} - q^{-96} +2 q^{-97} + q^{-98} + q^{-100} - q^{-103} - q^{-104} + q^{-105} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 42]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[18, 14, 1, 13], X[12, 18, 13, 17]]
In[3]:=	GaussCode[Knot[9, 42]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8]
In[4]:=	DTCode[Knot[9, 42]]
Out[4]=	DTCode[4, 8, 10, -14, 2, -16, -18, -6, -12]
In[5]:=	br = BR[Knot[9, 42]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 42]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 42]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 42]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[9, 42]][t]
Out[10]=	-2 2 2 -1 - t + - + 2 t - t t
In[11]:=	Conway[Knot[9, 42]][z]
Out[11]=	2 4 1 - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 42]}
In[13]:=	{KnotDet[Knot[9, 42]], KnotSignature[Knot[9, 42]]}
Out[13]=	{7, 2}
In[14]:=	Jones[Knot[9, 42]][q]
Out[14]=	-3 -2 1 2 3 -1 + q - q + - + q - q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 42]}
In[16]:=	A2Invariant[Knot[9, 42]][q]
Out[16]=	-10 -8 -6 -2 2 6 8 10 -1 + q + q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[9, 42]][a, z]
Out[17]=	2 2 2 2 z 2 2 4 -3 + -- + 2 a - 4 z + -- + a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 42]][a, z]
Out[18]=	2 3 2 2 2 z 2 6 z 2 2 6 z 3 -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + 6 a z - 2 a 2 a a a 4 5 6 7 4 5 z 2 4 5 z 5 6 z 2 6 z 7 10 z - ---- - 5 a z - ---- - 5 a z + 2 z + -- + a z + -- + a z 2 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[9, 42]], Vassiliev[3][Knot[9, 42]]}
Out[19]=	{-2, 0}
In[20]:=	Kh[Knot[9, 42]][q, t]
Out[20]=	1 3 1 1 1 1 q 3 7 2 - + q + q + ----- + ----- + ----- + --- + - + q t + q t q 7 4 3 3 3 2 q t t q t q t q t
In[21]:=	ColouredJones[Knot[9, 42], 2][q]
Out[21]=	-10 -9 -8 2 -6 -5 2 -3 1 3 4 -1 + q - q - q + -- - q - q + -- - q + - + q - q + 2 q - 7 4 q q q 5 6 7 8 9 10 q - q + 2 q - q - q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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