K11a305

K11a304

K11a306

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a305 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a305/ThurstonBennequinNumber Hyperbolic Volume 17.2512 A-Polynomial See Data:K11a305/A-polynomial

[edit Notes for K11a305's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a305's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 135, -2 }
Jones polynomial	$-q^{4}+4q^{3}-9q^{2}+14q-18+22q^{-1}-21q^{-2}+19q^{-3}-14q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+2z^{6}+3a^{4}z^{4}-9a^{2}z^{4}-z^{4}a^{-2}+7z^{4}+2a^{4}z^{2}-5a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}-a^{4}+a^{2}-a^{-2}+2$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+10a^{3}z^{9}+16az^{9}+6z^{9}a^{-1}+14a^{4}z^{8}+15a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+12a^{5}z^{7}-14a^{3}z^{7}-44az^{7}-17z^{7}a^{-1}+z^{7}a^{-3}+8a^{6}z^{6}-27a^{4}z^{6}-64a^{2}z^{6}-13z^{6}a^{-2}-42z^{6}+4a^{7}z^{5}-15a^{5}z^{5}-5a^{3}z^{5}+26az^{5}+9z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+20a^{4}z^{4}+64a^{2}z^{4}+13z^{4}a^{-2}+50z^{4}-2a^{7}z^{3}+4a^{5}z^{3}+9a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}+3z^{3}a^{-3}-6a^{4}z^{2}-19a^{2}z^{2}-6z^{2}a^{-2}-19z^{2}+a^{5}z-2az-2za^{-1}-za^{-3}-a^{4}-a^{2}+a^{-2}+2$
The A2 invariant	$q^{20}-2q^{18}+2q^{16}-3q^{14}-2q^{12}+3q^{10}-3q^{8}+6q^{6}-q^{4}+2q^{2}+2-3q^{-2}+3q^{-4}-2q^{-6}+q^{-10}-q^{-12}$
The G2 invariant	Data:K11a305/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a305 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["K11a305"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 135, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-9q^{2}+14q-18+22q^{-1}-21q^{-2}+19q^{-3}-14q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+2z^{6}+3a^{4}z^{4}-9a^{2}z^{4}-z^{4}a^{-2}+7z^{4}+2a^{4}z^{2}-5a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}-a^{4}+a^{2}-a^{-2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{10}+3z^{10}+10a^{3}z^{9}+16az^{9}+6z^{9}a^{-1}+14a^{4}z^{8}+15a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+12a^{5}z^{7}-14a^{3}z^{7}-44az^{7}-17z^{7}a^{-1}+z^{7}a^{-3}+8a^{6}z^{6}-27a^{4}z^{6}-64a^{2}z^{6}-13z^{6}a^{-2}-42z^{6}+4a^{7}z^{5}-15a^{5}z^{5}-5a^{3}z^{5}+26az^{5}+9z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+20a^{4}z^{4}+64a^{2}z^{4}+13z^{4}a^{-2}+50z^{4}-2a^{7}z^{3}+4a^{5}z^{3}+9a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}+3z^{3}a^{-3}-6a^{4}z^{2}-19a^{2}z^{2}-6z^{2}a^{-2}-19z^{2}+a^{5}z-2az-2za^{-1}-za^{-3}-a^{4}-a^{2}+a^{-2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a157, K11a264,}

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["K11a305"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $-q^{4}+4q^{3}-9q^{2}+14q-18+22q^{-1}-21q^{-2}+19q^{-3}-14q^{-4}+8q^{-5}-4q^{-6}+q^{-7}$ }

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]=

{K11a157, K11a264,}

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, -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 $8$ $-8$ $32$ ${\frac {172}{3}}$ ${\frac {44}{3}}$ $-64$ $-{\frac {368}{3}}$ $-{\frac {224}{3}}$ $24$ ${\frac {256}{3}}$ $32$ ${\frac {1376}{3}}$ ${\frac {352}{3}}$ ${\frac {8791}{15}}$ ${\frac {476}{15}}$ ${\frac {9604}{45}}$ ${\frac {425}{9}}$ ${\frac {871}{15}}$

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 K11a305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a304 K11a306