9 42

9_41

9_43

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42 at Knotilus!

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	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{11, 2}, {1, 9}, {10, 5}, {9, 11}, {8, 4}, {2, 7}, {6, 8}, {7, 10}, {5, 3}, {4, 1}, {3, 6}]

[edit Notes on presentations of 9 42]

Knot 9_42.

A graph, knot 9_42.

A part of a knot and a part of a graph.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 42"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8

In[6]:=

DTCode[K]

Out[6]=

4 8 10 -14 2 -16 -18 -6 -12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[22,3,2-]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(4,\{1,1,1,-2,-1,-1,3,-2,3\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 9, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{11, 2}, {1, 9}, {10, 5}, {9, 11}, {8, 4}, {2, 7}, {6, 8}, {7, 10}, {5, 3}, {4, 1}, {3, 6}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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 $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial	$-t^{2}+2t-1+2t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 2 }
Jones polynomial	$q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

Further Quantum Invariants

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	$q^{7}+q^{-7}$
2	$q^{22}-q^{18}+q^{6}+1+q^{-6}-q^{-18}+q^{-22}$
3	$q^{45}-q^{41}-q^{39}+q^{35}+q^{23}+q^{21}-q^{19}-q^{17}+q^{13}-q^{9}+q^{5}+q^{3}+q^{-3}+q^{-5}-q^{-13}-q^{-15}+q^{-21}+2q^{-23}-q^{-27}+2q^{-31}-3q^{-35}-q^{-37}+q^{-39}+q^{-41}$
4	$q^{76}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}-q^{58}+q^{48}+q^{46}-2q^{42}-2q^{40}+q^{36}+2q^{34}-q^{30}-q^{28}+2q^{24}+q^{22}+q^{20}-q^{18}-2q^{16}+q^{12}+2q^{10}-2q^{6}-q^{4}+2+q^{-2}-q^{-4}+q^{-8}+2q^{-10}+q^{-12}-q^{-14}+q^{-20}-q^{-24}-q^{-34}-2q^{-36}+q^{-40}+2q^{-42}+2q^{-44}-q^{-46}-q^{-48}-2q^{-50}+q^{-52}+3q^{-54}-2q^{-60}-q^{-62}+q^{-68}$
5	$q^{115}-q^{111}-q^{109}-q^{107}+q^{103}+2q^{101}+q^{99}-q^{95}-q^{93}-q^{91}+q^{87}+q^{81}+q^{79}-q^{75}-3q^{73}-2q^{71}+2q^{67}+3q^{65}+2q^{63}-2q^{59}-2q^{57}-q^{55}+q^{53}+2q^{51}+2q^{49}+q^{47}-q^{45}-2q^{43}-3q^{41}-2q^{39}+q^{37}+3q^{35}+3q^{33}+q^{31}-2q^{29}-4q^{27}-2q^{25}+q^{23}+3q^{21}+4q^{19}+q^{17}-2q^{15}-3q^{13}-q^{11}+2q^{9}+3q^{7}+2q^{5}-q^{3}-3q-2q^{-1}+q^{-3}+3q^{-5}+2q^{-7}-2q^{-11}-q^{-13}+q^{-15}+2q^{-17}+q^{-19}-q^{-21}-q^{-23}+q^{-27}-q^{-31}-q^{-33}+q^{-37}+q^{-39}+q^{-55}+2q^{-57}+q^{-59}-2q^{-61}-4q^{-63}-4q^{-65}+5q^{-69}+6q^{-71}+q^{-73}-5q^{-75}-6q^{-77}-3q^{-79}+3q^{-81}+6q^{-83}+5q^{-85}-3q^{-89}-3q^{-91}-2q^{-93}+q^{-97}+q^{-99}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
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

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} }

.

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]=

$-t^{2}+2t-1+2t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{4}-2z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 7, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$

"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}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 42"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

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} } , 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[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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 $-8$ $0$ $32$ ${\frac {164}{3}}$ ${\frac {76}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1312}{3}}$ $-{\frac {608}{3}}$ $-{\frac {6271}{15}}$ ${\frac {1484}{15}}$ $-{\frac {19564}{45}}$ ${\frac {607}{9}}$ $-{\frac {1471}{15}}$

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 $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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{10}-q^{9}-q^{8}+2q^{7}-q^{6}-q^{5}+2q^{4}-q^{3}+q-1+q^{-1}-q^{-3}+2q^{-4}-q^{-5}-q^{-6}+2q^{-7}-q^{-8}-q^{-9}+q^{-10}$
3	$q^{19}-2q^{17}-2q^{16}+4q^{15}+2q^{14}-4q^{13}-3q^{12}+5q^{11}+4q^{10}-5q^{9}-4q^{8}+5q^{7}+3q^{6}-5q^{5}-3q^{4}+5q^{3}+3q^{2}-4q-3+4q^{-1}+3q^{-2}-3q^{-3}-3q^{-4}+3q^{-5}+2q^{-6}-2q^{-7}-2q^{-8}+2q^{-9}+q^{-10}-2q^{-11}+2q^{-13}-2q^{-15}+2q^{-17}-q^{-19}-q^{-20}+q^{-21}$
4	$q^{32}-q^{31}-q^{29}-q^{28}+3q^{27}-q^{26}+3q^{25}-3q^{24}-4q^{23}+4q^{22}-q^{21}+6q^{20}-3q^{19}-5q^{18}+3q^{17}-3q^{16}+7q^{15}-2q^{14}-5q^{13}+3q^{12}-3q^{11}+6q^{10}-q^{9}-4q^{8}+2q^{7}-3q^{6}+5q^{5}+q^{4}-3q^{3}+q^{2}-4q+4+3q^{-1}-2q^{-2}-q^{-3}-5q^{-4}+3q^{-5}+5q^{-6}-2q^{-8}-6q^{-9}+q^{-10}+6q^{-11}+2q^{-12}-2q^{-13}-5q^{-14}-q^{-15}+5q^{-16}+2q^{-17}-q^{-18}-3q^{-19}-2q^{-20}+4q^{-21}-q^{-23}-q^{-24}-q^{-25}+4q^{-26}-q^{-27}-q^{-28}-q^{-29}-q^{-30}+3q^{-31}-q^{-34}-q^{-35}+q^{-36}$
5	$q^{47}-q^{45}-2q^{44}-q^{43}+4q^{41}+5q^{40}-5q^{38}-7q^{37}-3q^{36}+5q^{35}+11q^{34}+5q^{33}-6q^{32}-12q^{31}-7q^{30}+5q^{29}+13q^{28}+8q^{27}-5q^{26}-13q^{25}-8q^{24}+5q^{23}+13q^{22}+8q^{21}-5q^{20}-13q^{19}-8q^{18}+6q^{17}+13q^{16}+7q^{15}-6q^{14}-13q^{13}-7q^{12}+7q^{11}+12q^{10}+6q^{9}-6q^{8}-11q^{7}-6q^{6}+6q^{5}+10q^{4}+5q^{3}-4q^{2}-9q-5+4q^{-1}+7q^{-2}+4q^{-3}-2q^{-4}-6q^{-5}-4q^{-6}+3q^{-7}+4q^{-8}+2q^{-9}-q^{-10}-3q^{-11}-q^{-12}+2q^{-13}+2q^{-14}-q^{-15}-3q^{-16}-q^{-17}+2q^{-18}+4q^{-19}+2q^{-20}-3q^{-21}-6q^{-22}-2q^{-23}+3q^{-24}+5q^{-25}+4q^{-26}-2q^{-27}-6q^{-28}-3q^{-29}+q^{-30}+4q^{-31}+4q^{-32}-4q^{-34}-2q^{-35}+2q^{-37}+2q^{-38}-q^{-39}-2q^{-40}-q^{-41}+q^{-42}+2q^{-43}+q^{-44}-q^{-45}-q^{-46}-2q^{-47}+2q^{-49}+q^{-50}-q^{-53}-q^{-54}+q^{-55}$
6	$q^{66}-q^{65}-q^{62}-q^{61}-q^{60}+3q^{59}+q^{58}+4q^{57}+q^{56}-2q^{55}-7q^{54}-6q^{53}+2q^{51}+14q^{50}+7q^{49}+2q^{48}-13q^{47}-14q^{46}-7q^{45}-q^{44}+20q^{43}+13q^{42}+9q^{41}-12q^{40}-16q^{39}-13q^{38}-5q^{37}+20q^{36}+14q^{35}+13q^{34}-11q^{33}-14q^{32}-15q^{31}-7q^{30}+19q^{29}+13q^{28}+14q^{27}-11q^{26}-14q^{25}-15q^{24}-6q^{23}+19q^{22}+13q^{21}+14q^{20}-10q^{19}-14q^{18}-15q^{17}-5q^{16}+17q^{15}+11q^{14}+14q^{13}-8q^{12}-12q^{11}-15q^{10}-5q^{9}+13q^{8}+9q^{7}+16q^{6}-5q^{5}-9q^{4}-16q^{3}-6q^{2}+8q+7+18q^{-1}-5q^{-3}-17q^{-4}-8q^{-5}+2q^{-6}+4q^{-7}+19q^{-8}+5q^{-9}-16q^{-11}-8q^{-12}-4q^{-13}-q^{-14}+17q^{-15}+8q^{-16}+4q^{-17}-12q^{-18}-5q^{-19}-7q^{-20}-5q^{-21}+12q^{-22}+7q^{-23}+5q^{-24}-8q^{-25}-5q^{-27}-5q^{-28}+8q^{-29}+3q^{-30}+q^{-31}-7q^{-32}+2q^{-33}-q^{-34}-q^{-35}+8q^{-36}+2q^{-37}-3q^{-38}-8q^{-39}-q^{-40}-q^{-41}+q^{-42}+9q^{-43}+4q^{-44}-2q^{-45}-6q^{-46}-3q^{-47}-3q^{-48}-q^{-49}+8q^{-50}+3q^{-51}-2q^{-53}-2q^{-54}-2q^{-55}-2q^{-56}+6q^{-57}-q^{-59}-q^{-60}-q^{-61}-q^{-62}-q^{-63}+5q^{-64}-q^{-67}-q^{-68}-2q^{-69}-q^{-70}+3q^{-71}+q^{-73}-q^{-76}-q^{-77}+q^{-78}$
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, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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