10 132

10_131

10_133

Visit 10 132's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 132's page at Knotilus! Visit 10 132's page at the original Knot Atlas! 10 132 Quick Notes

10 132 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837
Gauss code	1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7
Dowker-Thistlethwaite code	4 8 -12 2 -16 -6 -20 -18 -10 -14
Conway Notation	[23,3,2-]

Minimum Braid Representative: Length is 11, 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 Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-1] Hyperbolic Volume 4.05686 A-Polynomial See Data:10 132/A-polynomial

[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated.

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 $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 10 132'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-t+1- t^{-1} + t^{-2} }
Conway polynomial	$z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 0 }
Jones polynomial	$q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+3a^{4}$
Kauffman polynomial (db, data sources)	$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-12a^{5}z^{5}-6a^{3}z^{5}+10a^{6}z^{4}+10a^{4}z^{4}+10a^{7}z^{3}+19a^{5}z^{3}+9a^{3}z^{3}-6a^{6}z^{2}-7a^{4}z^{2}-a^{2}z^{2}-5a^{7}z-8a^{5}z-4a^{3}z-az+2a^{6}+3a^{4}$
The A2 invariant	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
The G2 invariant	$q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2q^{80}-q^{78}-q^{76}-2q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2q^{52}+3q^{50}+q^{48}+q^{46}+2q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^{4}-q^{2}+1+q^{-6}-q^{-8}$

Further Quantum Invariants

Further quantum knot invariants for 10_132.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{7}+q^{5}+q^{3}$
2	$q^{44}-q^{40}-q^{30}-q^{24}+q^{16}+q^{14}+q^{10}+q^{6}+2-q^{-4}$
3	$-q^{87}+q^{83}+q^{81}-q^{77}+q^{67}-q^{63}+q^{59}+q^{57}-q^{55}-q^{53}-q^{45}-q^{43}+q^{39}-q^{35}+q^{31}-q^{27}-q^{25}+q^{17}+2q^{15}+2q^{13}+q^{7}+2q^{5}+q^{3}-q-q^{-1}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
1,1	$q^{60}+2q^{56}-2q^{50}-2q^{48}-2q^{44}+2q^{36}-2q^{34}-2q^{30}-q^{28}-2q^{24}+2q^{22}-q^{20}+4q^{18}+6q^{14}+q^{12}+4q^{10}+2q^{8}-2q^{6}+2q^{4}-2q^{2}+2-2q^{-2}$
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^{58}+q^{56}+q^{54}-q^{48}-q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{36}-q^{34}+q^{28}+q^{24}+2 q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{10}+q^8+q^4+q^2+1- q^{-2} - q^{-4} }

A3 Invariants.

Weight	Invariant
0,1,0	$q^{46}+q^{42}+q^{40}-2q^{36}-2q^{34}-4q^{32}-5q^{30}-2q^{28}-q^{26}+2q^{24}+3q^{22}+6q^{20}+4q^{18}+3q^{16}+2q^{14}+q^{12}-q^{10}-q^{8}-q^{4}+1$
1,0,0	$-q^{29}-q^{27}-2q^{25}-q^{23}+q^{19}+2q^{17}+2q^{15}+2q^{13}+q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{60}+q^{58}+2q^{56}+2q^{54}+2q^{52}+q^{50}-q^{48}-4q^{46}-6q^{44}-8q^{42}-9q^{40}-8q^{38}-5q^{36}+4q^{32}+8q^{30}+11q^{28}+11q^{26}+8q^{24}+6q^{22}+q^{20}-q^{18}-3q^{16}-2q^{14}-2q^{12}-q^{10}+q^{6}+1$
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^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}}

B2 Invariants.

Weight	Invariant
0,1	$-q^{46}-q^{42}-q^{40}+q^{30}+q^{26}+q^{22}+q^{16}+q^{12}+q^{10}+q^{8}+q^{4}-1$
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^{76}+q^{68}-q^{58}-2 q^{56}-q^{54}-q^{52}-2 q^{50}-2 q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{18}-q^{12}-q^4+1}

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^{62}+q^{58}+q^{56}+q^{54}-q^{50}-2 q^{48}-4 q^{46}-4 q^{44}-5 q^{42}-4 q^{40}-3 q^{38}+q^{34}+4 q^{32}+5 q^{30}+6 q^{28}+5 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}-q^{16}-q^{14}-q^{12}-q^8+1}

G2 Invariants.

Weight	Invariant
1,0	$q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2q^{80}-q^{78}-q^{76}-2q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2q^{52}+3q^{50}+q^{48}+q^{46}+2q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^{4}-q^{2}+1+q^{-6}-q^{-8}$

.

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["10 132"];

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+3z^{2}+1$

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]=

{ 5, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+3a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-6a^{4}z^{6}-6a^{7}z^{5}-12a^{5}z^{5}-6a^{3}z^{5}+10a^{6}z^{4}+10a^{4}z^{4}+10a^{7}z^{3}+19a^{5}z^{3}+9a^{3}z^{3}-6a^{6}z^{2}-7a^{4}z^{2}-a^{2}z^{2}-5a^{7}z-8a^{5}z-4a^{3}z-az+2a^{6}+3a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {5_1, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {5_1, ...}

Vassiliev invariants

V2 and V3:

(3, -5)

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 $12$ $-40$ $72$ $174$ $26$ $-480$ $-{\frac {2608}{3}}$ 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 -\frac{352}{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 -168} 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 288} 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 800} $2088$ $312$ 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 \frac{42751}{10}} $-{\frac {3506}{15}}$ ${\frac {10034}{5}}$ ${\frac {203}{2}}$ 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 \frac{2751}{10}}

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 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} ). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 10 132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -1             1 1 0 -3               1 1 -5         1 2     1 -7       1         1 -9       1 1       0 -11   1 1           0 -13                 0 -15 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$-q+1+2q^{-1}-3q^{-2}+q^{-3}+3q^{-4}-4q^{-5}+2q^{-6}+2q^{-7}-3q^{-8}+2q^{-9}+q^{-10}-3q^{-11}+2q^{-12}-2q^{-14}+2q^{-15}-q^{-16}-q^{-17}+2q^{-18}-q^{-19}-q^{-20}+q^{-21}$
3	$-q^{-1}+2q^{-3}+q^{-4}-2q^{-5}-q^{-6}+2q^{-7}+3q^{-8}-2q^{-9}-2q^{-10}+q^{-11}+3q^{-12}-2q^{-13}-3q^{-14}+q^{-15}+4q^{-16}-q^{-17}-4q^{-18}+5q^{-20}-5q^{-22}-q^{-23}+5q^{-24}+q^{-25}-5q^{-26}-q^{-27}+4q^{-28}+q^{-29}-3q^{-30}-q^{-31}+3q^{-32}-2q^{-34}+2q^{-36}-2q^{-38}+q^{-40}+q^{-41}-q^{-42}$
4	$q^{5}-q^{4}-q^{3}-q^{2}+5-q^{-2}-6q^{-3}-2q^{-4}+9q^{-5}+2q^{-6}-9q^{-8}-3q^{-9}+10q^{-10}+3q^{-11}+q^{-12}-10q^{-13}-3q^{-14}+11q^{-15}+3q^{-16}-q^{-17}-10q^{-18}-3q^{-19}+11q^{-20}+2q^{-21}-2q^{-22}-9q^{-23}-2q^{-24}+10q^{-25}+2q^{-26}-2q^{-27}-7q^{-28}-2q^{-29}+8q^{-30}+2q^{-31}-2q^{-32}-5q^{-33}-q^{-34}+6q^{-35}+q^{-36}-3q^{-37}-3q^{-38}+q^{-39}+5q^{-40}-5q^{-42}-2q^{-43}+2q^{-44}+6q^{-45}-6q^{-47}-2q^{-48}+q^{-49}+6q^{-50}+q^{-51}-4q^{-52}-2q^{-53}-q^{-54}+5q^{-55}-2q^{-57}-q^{-58}-q^{-59}+4q^{-60}-q^{-61}-q^{-62}-q^{-63}-q^{-64}+3q^{-65}-q^{-68}-q^{-69}+q^{-70}$
5	Not Available
6	Not Available
7	Not Available

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[10, 132]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19], X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], X[11, 6, 12, 7], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 132]]
Out[3]=	GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7]
In[4]:=	DTCode[Knot[10, 132]]
Out[4]=	DTCode[4, 8, -12, 2, -16, -6, -20, -18, -10, -14]
In[5]:=	br = BR[Knot[10, 132]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 132]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 132]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 132]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 132]][t]
Out[10]=	-2 1 2 1 + t - - - t + t t
In[11]:=	Conway[Knot[10, 132]][z]
Out[11]=	2 4 1 + 3 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[5, 1], Knot[10, 132]}
In[13]:=	{KnotDet[Knot[10, 132]], KnotSignature[Knot[10, 132]]}
Out[13]=	{5, 0}
In[14]:=	Jones[Knot[10, 132]][q]
Out[14]=	-7 -6 -5 -4 -2 -q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[5, 1], Knot[10, 132]}
In[16]:=	A2Invariant[Knot[10, 132]][q]
Out[16]=	-22 -20 -18 -14 -12 2 -8 -6 -q - q - q + q + q + --- + q + q 10 q
In[17]:=	HOMFLYPT[Knot[10, 132]][a, z]
Out[17]=	4 6 4 2 6 2 4 4 3 a - 2 a + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[10, 132]][a, z]
Out[18]=	4 6 3 5 7 2 2 4 2 3 a + 2 a - a z - 4 a z - 8 a z - 5 a z - a z - 7 a z - 6 2 3 3 5 3 7 3 4 4 6 4 6 a z + 9 a z + 19 a z + 10 a z + 10 a z + 10 a z - 3 5 5 5 7 5 4 6 6 6 3 7 5 7 6 a z - 12 a z - 6 a z - 6 a z - 6 a z + a z + 2 a z + 7 7 4 8 6 8 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 132]], Vassiliev[3][Knot[10, 132]]}
Out[19]=	{3, -5}
In[20]:=	Kh[Knot[10, 132]][q, t]
Out[20]=	-3 1 1 1 1 1 1 1 1 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 9 3 5 3 q t q t q t q t q t q t q t 2 1 ----- + --- 5 2 q t q t
In[21]:=	ColouredJones[Knot[10, 132], 2][q]
Out[21]=	-21 -20 -19 2 -17 -16 2 2 2 3 1 + q - q - q + --- - q - q + --- - --- + --- - --- + 18 15 14 12 11 q q q q q -10 2 3 2 2 4 3 -3 3 2 q + -- - -- + -- + -- - -- + -- + q - -- + - - q 9 8 7 6 5 4 2 q q q q q q q q

See/edit the Rolfsen_Splice_Template.