K11a307

K11a306

K11a308

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a307 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a307/ThurstonBennequinNumber Hyperbolic Volume 11.9838 A-Polynomial See Data:K11a307/A-polynomial

[edit Notes for K11a307'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 K11a307's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 83, -2 }
Jones polynomial	$-q^{2}+3q-5+9q^{-1}-11q^{-2}+13q^{-3}-13q^{-4}+11q^{-5}-8q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-3a^{6}+z^{6}a^{4}+3z^{4}a^{4}+4z^{2}a^{4}+2a^{4}+z^{6}a^{2}+3z^{4}a^{2}+3z^{2}a^{2}+a^{2}-z^{4}-2z^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+7z^{3}a^{9}-za^{9}+4z^{8}a^{8}-13z^{6}a^{8}+11z^{4}a^{8}-4z^{2}a^{8}+a^{8}+3z^{9}a^{7}-7z^{7}a^{7}+2z^{5}a^{7}+2z^{3}a^{7}-za^{7}+z^{10}a^{6}+4z^{8}a^{6}-22z^{6}a^{6}+33z^{4}a^{6}-18z^{2}a^{6}+3a^{6}+6z^{9}a^{5}-20z^{7}a^{5}+29z^{5}a^{5}-15z^{3}a^{5}+2za^{5}+z^{10}a^{4}+4z^{8}a^{4}-19z^{6}a^{4}+32z^{4}a^{4}-17z^{2}a^{4}+2a^{4}+3z^{9}a^{3}-6z^{7}a^{3}+8z^{5}a^{3}-5z^{3}a^{3}+2za^{3}+4z^{8}a^{2}-8z^{6}a^{2}+6z^{4}a^{2}-z^{2}a^{2}-a^{2}+4z^{7}a-8z^{5}a+3z^{3}a+3z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$
The A2 invariant	Data:K11a307/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a307/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a307 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["K11a307"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 83, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+3q-5+9q^{-1}-11q^{-2}+13q^{-3}-13q^{-4}+11q^{-5}-8q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-3a^{6}+z^{6}a^{4}+3z^{4}a^{4}+4z^{2}a^{4}+2a^{4}+z^{6}a^{2}+3z^{4}a^{2}+3z^{2}a^{2}+a^{2}-z^{4}-2z^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+7z^{3}a^{9}-za^{9}+4z^{8}a^{8}-13z^{6}a^{8}+11z^{4}a^{8}-4z^{2}a^{8}+a^{8}+3z^{9}a^{7}-7z^{7}a^{7}+2z^{5}a^{7}+2z^{3}a^{7}-za^{7}+z^{10}a^{6}+4z^{8}a^{6}-22z^{6}a^{6}+33z^{4}a^{6}-18z^{2}a^{6}+3a^{6}+6z^{9}a^{5}-20z^{7}a^{5}+29z^{5}a^{5}-15z^{3}a^{5}+2za^{5}+z^{10}a^{4}+4z^{8}a^{4}-19z^{6}a^{4}+32z^{4}a^{4}-17z^{2}a^{4}+2a^{4}+3z^{9}a^{3}-6z^{7}a^{3}+8z^{5}a^{3}-5z^{3}a^{3}+2za^{3}+4z^{8}a^{2}-8z^{6}a^{2}+6z^{4}a^{2}-z^{2}a^{2}-a^{2}+4z^{7}a-8z^{5}a+3z^{3}a+3z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_83, K11a323,}

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["K11a307"];

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]=

{ $2t^{3}-9t^{2}+19t-23+19t^{-1}-9t^{-2}+2t^{-3}$, $-q^{2}+3q-5+9q^{-1}-11q^{-2}+13q^{-3}-13q^{-4}+11q^{-5}-8q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$ }

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]=

{10_83, K11a323,}

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, 0)

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 $4$ $0$ $8$ $-{\frac {130}{3}}$ $-{\frac {62}{3}}$ $0$ $224$ $-32$ $160$ ${\frac {32}{3}}$ $0$ $-{\frac {520}{3}}$ $-{\frac {248}{3}}$ $-{\frac {23249}{30}}$ ${\frac {10178}{15}}$ $-{\frac {45538}{45}}$ $-{\frac {2863}{18}}$ $-{\frac {5009}{30}}$

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 K11a307. 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                   3 1   -2 -1                 6 2     4 -3               6 4       -2 -5             7 5         2 -7           6 6           0 -9         5 7             -2 -11       3 6               3 -13     2 5                 -3 -15   1 3                   2 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a306 K11a308