K11a349

K11a348

K11a350

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a349 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a349/ThurstonBennequinNumber Hyperbolic Volume 17.9314 A-Polynomial See Data:K11a349/A-polynomial

[edit Notes for K11a349'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 K11a349's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-14t^{2}+37t-49+37t^{-1}-14t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 155, 2 }
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-15q^{6}+21q^{5}-25q^{4}+25q^{3}-22q^{2}+17q-10+5q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-2}-2z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-4}+4z^{10}a^{-6}+10z^{9}a^{-3}+19z^{9}a^{-5}+9z^{9}a^{-7}+12z^{8}a^{-2}+9z^{8}a^{-4}+5z^{8}a^{-6}+8z^{8}a^{-8}+10z^{7}a^{-1}-12z^{7}a^{-3}-49z^{7}a^{-5}-23z^{7}a^{-7}+4z^{7}a^{-9}-15z^{6}a^{-2}-30z^{6}a^{-4}-32z^{6}a^{-6}-21z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}+49z^{5}a^{-5}+22z^{5}a^{-7}-9z^{5}a^{-9}+21z^{4}a^{-4}+35z^{4}a^{-6}+17z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}+2z^{3}a^{-1}-4z^{3}a^{-3}-19z^{3}a^{-5}-10z^{3}a^{-7}+3z^{3}a^{-9}+2z^{2}a^{-2}-8z^{2}a^{-6}-6z^{2}a^{-8}+za^{-3}+3za^{-5}+2za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	Data:K11a349/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a349/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a349 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["K11a349"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-14t^{2}+37t-49+37t^{-1}-14t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 155, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-4q^{8}+9q^{7}-15q^{6}+21q^{5}-25q^{4}+25q^{3}-22q^{2}+17q-10+5q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-2}-2z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}-2a^{-4}+a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-4}+4z^{10}a^{-6}+10z^{9}a^{-3}+19z^{9}a^{-5}+9z^{9}a^{-7}+12z^{8}a^{-2}+9z^{8}a^{-4}+5z^{8}a^{-6}+8z^{8}a^{-8}+10z^{7}a^{-1}-12z^{7}a^{-3}-49z^{7}a^{-5}-23z^{7}a^{-7}+4z^{7}a^{-9}-15z^{6}a^{-2}-30z^{6}a^{-4}-32z^{6}a^{-6}-21z^{6}a^{-8}+z^{6}a^{-10}+5z^{6}+az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}+49z^{5}a^{-5}+22z^{5}a^{-7}-9z^{5}a^{-9}+21z^{4}a^{-4}+35z^{4}a^{-6}+17z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}+2z^{3}a^{-1}-4z^{3}a^{-3}-19z^{3}a^{-5}-10z^{3}a^{-7}+3z^{3}a^{-9}+2z^{2}a^{-2}-8z^{2}a^{-6}-6z^{2}a^{-8}+za^{-3}+3za^{-5}+2za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a349"];

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]=

{ $2t^{3}-14t^{2}+37t-49+37t^{-1}-14t^{-2}+2t^{-3}$, $q^{9}-4q^{8}+9q^{7}-15q^{6}+21q^{5}-25q^{4}+25q^{3}-22q^{2}+17q-10+5q^{-1}-q^{-2}$ }

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:

(-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 $-4$ $-16$ $8$ $-{\frac {14}{3}}$ ${\frac {62}{3}}$ $64$ ${\frac {416}{3}}$ ${\frac {320}{3}}$ $16$ $-{\frac {32}{3}}$ $128$ ${\frac {56}{3}}$ $-{\frac {248}{3}}$ ${\frac {14849}{30}}$ ${\frac {4702}{15}}$ ${\frac {178}{45}}$ $-{\frac {161}{18}}$ $-{\frac {991}{30}}$

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 K11a349. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 9 3     -6 11               12 6       6 9             13 9         -4 7           12 12           0 5         10 13             3 3       7 12               -5 1     4 11                 7 -1   1 6                   -5 -3   4                     4 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a348 K11a350