K11a331

K11a330

K11a332

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a331 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X18,5,19,6 X22,8,1,7 X16,9,17,10 X4,12,5,11 X20,13,21,14 X10,15,11,16 X8,17,9,18 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, 2, -6, 3, -1, 4, -9, 5, -8, 6, -2, 7, -11, 8, -5, 9, -3, 10, -7, 11, -4
Dowker-Thistlethwaite code	6 12 18 22 16 4 20 10 8 2 14

A Braid Representative	{{{braid_table}}}
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:K11a331/ThurstonBennequinNumber Hyperbolic Volume 16.0068 A-Polynomial See Data:K11a331/A-polynomial

[edit Notes for K11a331'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 K11a331's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, -2 }
Jones polynomial	$q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{6}a^{4}+3z^{4}a^{4}+6z^{2}a^{4}+4a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-2z^{4}-3z^{2}-1+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+10a^{3}z^{9}+4az^{9}+8a^{6}z^{8}+7a^{4}z^{8}+3a^{2}z^{8}+4z^{8}+6a^{7}z^{7}-11a^{5}z^{7}-25a^{3}z^{7}-5az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-21a^{6}z^{6}-28a^{4}z^{6}-11a^{2}z^{6}+z^{6}a^{-2}-6z^{6}+a^{9}z^{5}-13a^{7}z^{5}+7a^{5}z^{5}+28a^{3}z^{5}-az^{5}-8z^{5}a^{-1}-5a^{8}z^{4}+26a^{6}z^{4}+39a^{4}z^{4}+7a^{2}z^{4}-3z^{4}a^{-2}-4z^{4}-2a^{9}z^{3}+10a^{7}z^{3}+6a^{5}z^{3}-14a^{3}z^{3}-2az^{3}+6z^{3}a^{-1}-12a^{6}z^{2}-20a^{4}z^{2}-5a^{2}z^{2}+3z^{2}a^{-2}+6z^{2}-3a^{7}z-3a^{5}z+a^{3}z-za^{-1}+2a^{6}+4a^{4}+a^{2}-a^{-2}-1$
The A2 invariant	Data:K11a331/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a331/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a331 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["K11a331"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$

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]=

{ 115, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{6}a^{4}+3z^{4}a^{4}+6z^{2}a^{4}+4a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-2z^{4}-3z^{2}-1+z^{2}a^{-2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+10a^{3}z^{9}+4az^{9}+8a^{6}z^{8}+7a^{4}z^{8}+3a^{2}z^{8}+4z^{8}+6a^{7}z^{7}-11a^{5}z^{7}-25a^{3}z^{7}-5az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-21a^{6}z^{6}-28a^{4}z^{6}-11a^{2}z^{6}+z^{6}a^{-2}-6z^{6}+a^{9}z^{5}-13a^{7}z^{5}+7a^{5}z^{5}+28a^{3}z^{5}-az^{5}-8z^{5}a^{-1}-5a^{8}z^{4}+26a^{6}z^{4}+39a^{4}z^{4}+7a^{2}z^{4}-3z^{4}a^{-2}-4z^{4}-2a^{9}z^{3}+10a^{7}z^{3}+6a^{5}z^{3}-14a^{3}z^{3}-2az^{3}+6z^{3}a^{-1}-12a^{6}z^{2}-20a^{4}z^{2}-5a^{2}z^{2}+3z^{2}a^{-2}+6z^{2}-3a^{7}z-3a^{5}z+a^{3}z-za^{-1}+2a^{6}+4a^{4}+a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a41, K11a183, K11a198,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a3, K11a51,}

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["K11a331"];

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]=

{ $2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$, $q^{3}-3q^{2}+7q-12+16q^{-1}-18q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{10_121, K11a41, K11a183, K11a198,}

In[6]:=

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

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

Out[6]=

{K11a3, K11a51,}

Vassiliev invariants

V2 and V3:

(1, -4)

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 $4$ $-32$ $8$ ${\frac {206}{3}}$ $-{\frac {14}{3}}$ $-128$ $-{\frac {896}{3}}$ ${\frac {256}{3}}$ $-160$ ${\frac {32}{3}}$ $512$ ${\frac {824}{3}}$ $-{\frac {56}{3}}$ ${\frac {45631}{30}}$ $-{\frac {2954}{5}}$ ${\frac {49982}{45}}$ ${\frac {1793}{18}}$ ${\frac {4351}{30}}$

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 K11a331. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   5 1   4 1                 7 2     -5 -1               9 5       4 -3             10 8         -2 -5           9 8           1 -7         7 10             3 -9       5 9               -4 -11     2 7                 5 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a330 K11a332