K11n131

K11n130

K11n132

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n131 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X18,5,19,6 X7,16,8,17 X9,14,10,15 X2,11,3,12 X20,14,21,13 X15,8,16,9 X22,17,1,18 X12,20,13,19 X6,21,7,22
Gauss code	1, -6, 2, -1, 3, -11, -4, 8, -5, -2, 6, -10, 7, 5, -8, 4, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 18 -16 -14 2 20 -8 22 12 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n131/ThurstonBennequinNumber Hyperbolic Volume 14.3723 A-Polynomial See Data:K11n131/A-polynomial

[edit Notes for K11n131's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11n131's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, -2 }
Jones polynomial	$-q^{2}+4q-7+10q^{-1}-11q^{-2}+12q^{-3}-10q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}-2z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+2a^{2}-z^{4}-z^{2}$
Kauffman polynomial (db, data sources)	$a^{5}z^{9}+a^{3}z^{9}+a^{6}z^{8}+5a^{4}z^{8}+4a^{2}z^{8}+a^{5}z^{7}+7a^{3}z^{7}+6az^{7}+a^{6}z^{6}-5a^{4}z^{6}-2a^{2}z^{6}+4z^{6}+4a^{7}z^{5}+2a^{5}z^{5}-14a^{3}z^{5}-11az^{5}+z^{5}a^{-1}+a^{8}z^{4}+a^{6}z^{4}-a^{4}z^{4}-8a^{2}z^{4}-7z^{4}-4a^{7}z^{3}-3a^{5}z^{3}+5a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+4a^{4}z^{2}+6a^{2}z^{2}+2z^{2}+a^{5}z+a^{3}z-a^{4}-2a^{2}$
The A2 invariant	Data:K11n131/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n131/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n131 Quantum Invariants.

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["K11n131"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 67, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+4q-7+10q^{-1}-11q^{-2}+12q^{-3}-10q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{9}+a^{3}z^{9}+a^{6}z^{8}+5a^{4}z^{8}+4a^{2}z^{8}+a^{5}z^{7}+7a^{3}z^{7}+6az^{7}+a^{6}z^{6}-5a^{4}z^{6}-2a^{2}z^{6}+4z^{6}+4a^{7}z^{5}+2a^{5}z^{5}-14a^{3}z^{5}-11az^{5}+z^{5}a^{-1}+a^{8}z^{4}+a^{6}z^{4}-a^{4}z^{4}-8a^{2}z^{4}-7z^{4}-4a^{7}z^{3}-3a^{5}z^{3}+5a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+4a^{4}z^{2}+6a^{2}z^{2}+2z^{2}+a^{5}z+a^{3}z-a^{4}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n7, K11n160,}

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

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["K11n131"];

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]=

{ $t^{3}-6t^{2}+16t-21+16t^{-1}-6t^{-2}+t^{-3}$, $-q^{2}+4q-7+10q^{-1}-11q^{-2}+12q^{-3}-10q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$ }

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]=

{K11n7, K11n160,}

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:

(1, -1)

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 $4$ $-8$ $8$ ${\frac {62}{3}}$ ${\frac {10}{3}}$ $-32$ $-{\frac {176}{3}}$ $-{\frac {128}{3}}$ $24$ ${\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {40}{3}}$ ${\frac {5071}{30}}$ ${\frac {2938}{15}}$ $-{\frac {2698}{45}}$ $-{\frac {1135}{18}}$ $-{\frac {689}{30}}$

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 K11n131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11n130 K11n132