K11a312

K11a311

K11a313

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a312 at Knotilus!

Knot presentations

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

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:K11a312/ThurstonBennequinNumber Hyperbolic Volume 16.0409 A-Polynomial See Data:K11a312/A-polynomial

[edit Notes for K11a312'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 K11a312's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-13t^{2}+27t-33+27t^{-1}-13t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+5z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$-q^{2}+3q-6+12q^{-1}-16q^{-2}+19q^{-3}-19q^{-4}+17q^{-5}-13q^{-6}+8q^{-7}-4q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-3z^{4}a^{6}-7z^{2}a^{6}-4a^{6}+2z^{6}a^{4}+7z^{4}a^{4}+9z^{2}a^{4}+4a^{4}+z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-z^{4}-2z^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-10z^{5}a^{9}+6z^{3}a^{9}+6z^{8}a^{8}-13z^{6}a^{8}+5z^{4}a^{8}-z^{2}a^{8}+a^{8}+5z^{9}a^{7}-6z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+2z^{10}a^{6}+7z^{8}a^{6}-27z^{6}a^{6}+29z^{4}a^{6}-18z^{2}a^{6}+4a^{6}+11z^{9}a^{5}-28z^{7}a^{5}+31z^{5}a^{5}-16z^{3}a^{5}+2za^{5}+2z^{10}a^{4}+8z^{8}a^{4}-33z^{6}a^{4}+48z^{4}a^{4}-25z^{2}a^{4}+4a^{4}+6z^{9}a^{3}-13z^{7}a^{3}+16z^{5}a^{3}-6z^{3}a^{3}+2za^{3}+7z^{8}a^{2}-17z^{6}a^{2}+20z^{4}a^{2}-7z^{2}a^{2}+5z^{7}a-9z^{5}a+4z^{3}a+3z^{6}-6z^{4}+2z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$
The A2 invariant	Data:K11a312/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a312/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a312 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["K11a312"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-13t^{2}+27t-33+27t^{-1}-13t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+5z^{4}+2z^{2}+1$

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]=

{ 119, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+3q-6+12q^{-1}-16q^{-2}+19q^{-3}-19q^{-4}+17q^{-5}-13q^{-6}+8q^{-7}-4q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+a^{8}-3z^{4}a^{6}-7z^{2}a^{6}-4a^{6}+2z^{6}a^{4}+7z^{4}a^{4}+9z^{2}a^{4}+4a^{4}+z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-z^{4}-2z^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-10z^{5}a^{9}+6z^{3}a^{9}+6z^{8}a^{8}-13z^{6}a^{8}+5z^{4}a^{8}-z^{2}a^{8}+a^{8}+5z^{9}a^{7}-6z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+2z^{10}a^{6}+7z^{8}a^{6}-27z^{6}a^{6}+29z^{4}a^{6}-18z^{2}a^{6}+4a^{6}+11z^{9}a^{5}-28z^{7}a^{5}+31z^{5}a^{5}-16z^{3}a^{5}+2za^{5}+2z^{10}a^{4}+8z^{8}a^{4}-33z^{6}a^{4}+48z^{4}a^{4}-25z^{2}a^{4}+4a^{4}+6z^{9}a^{3}-13z^{7}a^{3}+16z^{5}a^{3}-6z^{3}a^{3}+2za^{3}+7z^{8}a^{2}-17z^{6}a^{2}+20z^{4}a^{2}-7z^{2}a^{2}+5z^{7}a-9z^{5}a+4z^{3}a+3z^{6}-6z^{4}+2z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$

"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["K11a312"];

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]=

{ $3t^{3}-13t^{2}+27t-33+27t^{-1}-13t^{-2}+3t^{-3}$, $-q^{2}+3q-6+12q^{-1}-16q^{-2}+19q^{-3}-19q^{-4}+17q^{-5}-13q^{-6}+8q^{-7}-4q^{-8}+q^{-9}$ }

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:

(2, -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 $8$ $-16$ $32$ ${\frac {28}{3}}$ $-{\frac {76}{3}}$ $-128$ ${\frac {224}{3}}$ $-{\frac {160}{3}}$ $208$ ${\frac {256}{3}}$ $128$ ${\frac {224}{3}}$ $-{\frac {608}{3}}$ $-{\frac {7649}{15}}$ ${\frac {6332}{5}}$ $-{\frac {78356}{45}}$ $-{\frac {2047}{9}}$ $-{\frac {5249}{15}}$

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 K11a312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a311 K11a313