K11a232

K11a231

K11a233

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a232 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X14,4,15,3 X10,5,11,6 X16,8,17,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X20,15,21,16 X6,18,7,17 X12,19,13,20 X8,21,9,22
Gauss code	1, -7, 2, -1, 3, -9, 4, -11, 5, -3, 6, -10, 7, -2, 8, -4, 9, -6, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 14 10 16 22 18 2 20 6 12 8

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a232/ThurstonBennequinNumber Hyperbolic Volume 16.4523 A-Polynomial See Data:K11a232/A-polynomial

[edit Notes for K11a232's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a232's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-14t^{2}+26t-31+26t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-4z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, -2 }
Jones polynomial	$-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-19q^{-2}+17q^{-3}-13q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+6a^{4}z^{2}-16a^{2}z^{2}-2z^{2}a^{-2}+11z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+12az^{9}+5z^{9}a^{-1}+11a^{4}z^{8}+15a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+10a^{5}z^{7}-3a^{3}z^{7}-27az^{7}-13z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-21a^{4}z^{6}-56a^{2}z^{6}-14z^{6}a^{-2}-43z^{6}+3a^{7}z^{5}-17a^{5}z^{5}-24a^{3}z^{5}+2az^{5}+3z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+18a^{4}z^{4}+58a^{2}z^{4}+15z^{4}a^{-2}+49z^{4}-2a^{7}z^{3}+16a^{5}z^{3}+29a^{3}z^{3}+17az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-8a^{4}z^{2}-29a^{2}z^{2}-5z^{2}a^{-2}-23z^{2}-4a^{5}z-9a^{3}z-8az-4za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	$q^{20}-q^{18}+3q^{16}-2q^{14}-2q^{12}+2q^{10}-4q^{8}+4q^{6}-2q^{4}+q^{2}+2-2q^{-2}+4q^{-4}-q^{-6}+q^{-10}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+6q^{106}-5q^{104}+8q^{100}-17q^{98}+27q^{96}-36q^{94}+35q^{92}-24q^{90}+2q^{88}+34q^{86}-69q^{84}+102q^{82}-125q^{80}+113q^{78}-69q^{76}-18q^{74}+134q^{72}-233q^{70}+303q^{68}-286q^{66}+170q^{64}+31q^{62}-264q^{60}+441q^{58}-476q^{56}+329q^{54}-59q^{52}-238q^{50}+451q^{48}-459q^{46}+263q^{44}+53q^{42}-355q^{40}+479q^{38}-373q^{36}+46q^{34}+338q^{32}-602q^{30}+642q^{28}-417q^{26}+7q^{24}+423q^{22}-723q^{20}+769q^{18}-561q^{16}+159q^{14}+279q^{12}-582q^{10}+674q^{8}-505q^{6}+172q^{4}+194q^{2}-455+489q^{-2}-291q^{-4}-46q^{-6}+377q^{-8}-528q^{-10}+446q^{-12}-151q^{-14}-218q^{-16}+509q^{-18}-603q^{-20}+468q^{-22}-178q^{-24}-151q^{-26}+385q^{-28}-448q^{-30}+361q^{-32}-175q^{-34}-14q^{-36}+139q^{-38}-187q^{-40}+156q^{-42}-91q^{-44}+29q^{-46}+15q^{-48}-32q^{-50}+28q^{-52}-19q^{-54}+9q^{-56}-3q^{-58}+q^{-60}$

Further Quantum Invariants

K11a232 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["K11a232"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-14t^{2}+26t-31+26t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-4z^{4}-z^{2}+1$

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]=

{ 123, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-19q^{-2}+17q^{-3}-13q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+6a^{4}z^{2}-16a^{2}z^{2}-2z^{2}a^{-2}+11z^{2}+2a^{4}-5a^{2}-a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+12az^{9}+5z^{9}a^{-1}+11a^{4}z^{8}+15a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+10a^{5}z^{7}-3a^{3}z^{7}-27az^{7}-13z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-21a^{4}z^{6}-56a^{2}z^{6}-14z^{6}a^{-2}-43z^{6}+3a^{7}z^{5}-17a^{5}z^{5}-24a^{3}z^{5}+2az^{5}+3z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+18a^{4}z^{4}+58a^{2}z^{4}+15z^{4}a^{-2}+49z^{4}-2a^{7}z^{3}+16a^{5}z^{3}+29a^{3}z^{3}+17az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-8a^{4}z^{2}-29a^{2}z^{2}-5z^{2}a^{-2}-23z^{2}-4a^{5}z-9a^{3}z-8az-4za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"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["K11a232"];

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]=

{ $-t^{4}+5t^{3}-14t^{2}+26t-31+26t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+4q^{3}-8q^{2}+13q-17+20q^{-1}-19q^{-2}+17q^{-3}-13q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{}

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 $-4$ $16$ $8$ ${\frac {82}{3}}$ ${\frac {110}{3}}$ $-64$ $-{\frac {608}{3}}$ $-{\frac {320}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ $-{\frac {328}{3}}$ $-{\frac {440}{3}}$ ${\frac {9569}{30}}$ ${\frac {6622}{15}}$ $-{\frac {17342}{45}}$ ${\frac {1759}{18}}$ $-{\frac {4831}{30}}$

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 K11a232. 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                   5 1   -4 3                 8 3     5 1               9 5       -4 -1             11 8         3 -3           9 10           1 -5         8 10             -2 -7       5 9               4 -9     2 8                 -6 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a231 K11a233