K11a249

K11a248

K11a250

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a249 at Knotilus!

Knot presentations

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

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:K11a249/ThurstonBennequinNumber Hyperbolic Volume 15.797 A-Polynomial See Data:K11a249/A-polynomial

[edit Notes for K11a249's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,3]$ Rasmussen s-Invariant 0

[edit Notes for K11a249's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+11t^{2}-27t+37-27t^{-1}+11t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 117, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-12q+17-18q^{-1}+18q^{-2}-16q^{-3}+11q^{-4}-7q^{-5}+4q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}-z^{6}a^{2}-2z^{4}a^{2}-3z^{2}a^{2}-2a^{2}-z^{6}-2z^{4}-z^{2}+2+z^{4}a^{-2}+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+6a^{5}z^{9}+14a^{3}z^{9}+8az^{9}+4a^{6}z^{8}-a^{4}z^{8}+5a^{2}z^{8}+10z^{8}+a^{7}z^{7}-23a^{5}z^{7}-49a^{3}z^{7}-15az^{7}+10z^{7}a^{-1}-15a^{6}z^{6}-26a^{4}z^{6}-33a^{2}z^{6}+8z^{6}a^{-2}-14z^{6}-3a^{7}z^{5}+25a^{5}z^{5}+53a^{3}z^{5}+10az^{5}-11z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+37a^{4}z^{4}+35a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}+a^{7}z^{3}-8a^{5}z^{3}-18a^{3}z^{3}-6az^{3}+z^{3}a^{-1}-2z^{3}a^{-3}-3a^{6}z^{2}-12a^{4}z^{2}-12a^{2}z^{2}+2z^{2}a^{-2}-z^{2}-a^{5}z-a^{3}z+a^{4}+2a^{2}+2$
The A2 invariant	Data:K11a249/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a249/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a249 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["K11a249"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+11t^{2}-27t+37-27t^{-1}+11t^{-2}-2t^{-3}$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 117, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-4q^{3}+8q^{2}-12q+17-18q^{-1}+18q^{-2}-16q^{-3}+11q^{-4}-7q^{-5}+4q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+6a^{5}z^{9}+14a^{3}z^{9}+8az^{9}+4a^{6}z^{8}-a^{4}z^{8}+5a^{2}z^{8}+10z^{8}+a^{7}z^{7}-23a^{5}z^{7}-49a^{3}z^{7}-15az^{7}+10z^{7}a^{-1}-15a^{6}z^{6}-26a^{4}z^{6}-33a^{2}z^{6}+8z^{6}a^{-2}-14z^{6}-3a^{7}z^{5}+25a^{5}z^{5}+53a^{3}z^{5}+10az^{5}-11z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+37a^{4}z^{4}+35a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}+a^{7}z^{3}-8a^{5}z^{3}-18a^{3}z^{3}-6az^{3}+z^{3}a^{-1}-2z^{3}a^{-3}-3a^{6}z^{2}-12a^{4}z^{2}-12a^{2}z^{2}+2z^{2}a^{-2}-z^{2}-a^{5}z-a^{3}z+a^{4}+2a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a8, K11a38, K11a187,}

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["K11a249"];

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]=

{ $-2t^{3}+11t^{2}-27t+37-27t^{-1}+11t^{-2}-2t^{-3}$, $q^{4}-4q^{3}+8q^{2}-12q+17-18q^{-1}+18q^{-2}-16q^{-3}+11q^{-4}-7q^{-5}+4q^{-6}-q^{-7}$ }

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]=

{K11a8, K11a38, K11a187,}

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 $-4$ $8$ $8$ ${\frac {34}{3}}$ ${\frac {38}{3}}$ $-32$ $-{\frac {208}{3}}$ ${\frac {128}{3}}$ $-88$ $-{\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ ${\frac {2129}{30}}$ $-{\frac {2698}{15}}$ ${\frac {3898}{45}}$ ${\frac {2191}{18}}$ ${\frac {209}{30}}$

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 K11a249. 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 9                       1 1 7                     3   -3 5                   5 1   4 3                 7 3     -4 1               10 5       5 -1             9 8         -1 -3           9 9           0 -5         7 9             2 -7       4 9               -5 -9     3 7                 4 -11   1 4                   -3 -13   3                     3 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a248 K11a250