K11a265

K11a264

K11a266

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a265 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X20,12,21,11 X18,14,19,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code	1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code	6 8 16 14 4 20 18 2 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a265/ThurstonBennequinNumber Hyperbolic Volume 14.8555 A-Polynomial See Data:K11a265/A-polynomial

[edit Notes for K11a265's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a265's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-25t+33-25t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 109, 0 }
Jones polynomial	$-q^{7}+3q^{6}-6q^{5}+10q^{4}-14q^{3}+17q^{2}-17q+16-12q^{-1}+8q^{-2}-4q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-a^{-2}+2a^{-4}-a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+10z^{9}a^{-3}+4z^{9}a^{-5}+7z^{8}a^{-2}+z^{8}a^{-4}+3z^{8}a^{-6}+9z^{8}+10az^{7}-6z^{7}a^{-1}-31z^{7}a^{-3}-14z^{7}a^{-5}+z^{7}a^{-7}+8a^{2}z^{6}-29z^{6}a^{-2}-22z^{6}a^{-4}-12z^{6}a^{-6}-11z^{6}+4a^{3}z^{5}-12az^{5}-3z^{5}a^{-1}+30z^{5}a^{-3}+13z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+27z^{4}a^{-2}+31z^{4}a^{-4}+14z^{4}a^{-6}+z^{4}-2a^{3}z^{3}+3az^{3}-12z^{3}a^{-3}-3z^{3}a^{-5}+4z^{3}a^{-7}+2a^{2}z^{2}-10z^{2}a^{-2}-14z^{2}a^{-4}-6z^{2}a^{-6}+za^{-1}+2za^{-3}-za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$
The A2 invariant	Data:K11a265/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a265/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a265 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["K11a265"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-25t+33-25t^{-1}+11t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-z^{4}+z^{2}+1$

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]=

{ 109, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+3q^{6}-6q^{5}+10q^{4}-14q^{3}+17q^{2}-17q+16-12q^{-1}+8q^{-2}-4q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-a^{-2}+2a^{-4}-a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+10z^{9}a^{-3}+4z^{9}a^{-5}+7z^{8}a^{-2}+z^{8}a^{-4}+3z^{8}a^{-6}+9z^{8}+10az^{7}-6z^{7}a^{-1}-31z^{7}a^{-3}-14z^{7}a^{-5}+z^{7}a^{-7}+8a^{2}z^{6}-29z^{6}a^{-2}-22z^{6}a^{-4}-12z^{6}a^{-6}-11z^{6}+4a^{3}z^{5}-12az^{5}-3z^{5}a^{-1}+30z^{5}a^{-3}+13z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-8a^{2}z^{4}+27z^{4}a^{-2}+31z^{4}a^{-4}+14z^{4}a^{-6}+z^{4}-2a^{3}z^{3}+3az^{3}-12z^{3}a^{-3}-3z^{3}a^{-5}+4z^{3}a^{-7}+2a^{2}z^{2}-10z^{2}a^{-2}-14z^{2}a^{-4}-6z^{2}a^{-6}+za^{-1}+2za^{-3}-za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a56, K11a185,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a185,}

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["K11a265"];

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]=

{ $-2t^{3}+11t^{2}-25t+33-25t^{-1}+11t^{-2}-2t^{-3}$, $-q^{7}+3q^{6}-6q^{5}+10q^{4}-14q^{3}+17q^{2}-17q+16-12q^{-1}+8q^{-2}-4q^{-3}+q^{-4}$ }

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]=

{K11a56, K11a185,}

In[6]:=

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

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

Out[6]=

{K11a185,}

Vassiliev invariants

V2 and V3:

(1, 2)

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 $4$ $16$ $8$ ${\frac {158}{3}}$ ${\frac {34}{3}}$ $64$ ${\frac {544}{3}}$ $-{\frac {128}{3}}$ $112$ ${\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {136}{3}}$ ${\frac {19471}{30}}$ $-{\frac {1594}{5}}$ ${\frac {23102}{45}}$ ${\frac {2129}{18}}$ ${\frac {2671}{30}}$

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=$0 is the signature of K11a265. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   4 1   -3 9                 6 2     4 7               8 4       -4 5             9 6         3 3           8 8           0 1         8 9             -1 -1       5 9               4 -3     3 7                 -4 -5   1 5                   4 -7   3                     -3 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a264 K11a266