K11a231

K11a230

K11a232

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a231 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a231/ThurstonBennequinNumber Hyperbolic Volume 14.7354 A-Polynomial See Data:K11a231/A-polynomial

[edit Notes for K11a231's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a231's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 99, -2 }
Jones polynomial	$-q^{4}+2q^{3}-5q^{2}+10q-12+16q^{-1}-16q^{-2}+14q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+10z^{4}+6a^{4}z^{2}-19a^{2}z^{2}-4z^{2}a^{-2}+18z^{2}+3a^{4}-10a^{2}-4a^{-2}+12$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+6az^{9}+2z^{9}a^{-1}+8a^{4}z^{8}+10a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+9a^{5}z^{7}+4a^{3}z^{7}-9az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-12a^{4}z^{6}-31a^{2}z^{6}-8z^{6}a^{-2}-21z^{6}+3a^{7}z^{5}-16a^{5}z^{5}-29a^{3}z^{5}-14az^{5}-9z^{5}a^{-1}-5z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+6a^{4}z^{4}+30a^{2}z^{4}+11z^{4}a^{-2}+28z^{4}-2a^{7}z^{3}+16a^{5}z^{3}+35a^{3}z^{3}+29az^{3}+20z^{3}a^{-1}+8z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-2a^{4}z^{2}-22a^{2}z^{2}-8z^{2}a^{-2}-24z^{2}-6a^{5}z-16a^{3}z-17az-11za^{-1}-4za^{-3}+3a^{4}+10a^{2}+4a^{-2}+12$
The A2 invariant	$q^{20}-q^{18}+3q^{16}-q^{14}-q^{12}-6q^{8}+q^{6}-3q^{4}+4q^{2}+5+2q^{-2}+4q^{-4}-2q^{-6}-q^{-8}-q^{-10}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+6q^{106}-5q^{104}+9q^{100}-19q^{98}+29q^{96}-38q^{94}+35q^{92}-22q^{90}-5q^{88}+49q^{86}-87q^{84}+115q^{82}-119q^{80}+82q^{78}-16q^{76}-81q^{74}+178q^{72}-232q^{70}+223q^{68}-133q^{66}-4q^{64}+156q^{62}-252q^{60}+268q^{58}-177q^{56}+20q^{54}+136q^{52}-227q^{50}+204q^{48}-65q^{46}-111q^{44}+242q^{42}-274q^{40}+170q^{38}+19q^{36}-243q^{34}+375q^{32}-397q^{30}+265q^{28}-43q^{26}-210q^{24}+377q^{22}-414q^{20}+315q^{18}-126q^{16}-99q^{14}+262q^{12}-302q^{10}+229q^{8}-54q^{6}-119q^{4}+242q^{2}-219+99q^{-2}+86q^{-4}-233q^{-6}+304q^{-8}-242q^{-10}+84q^{-12}+106q^{-14}-243q^{-16}+304q^{-18}-249q^{-20}+121q^{-22}+17q^{-24}-131q^{-26}+174q^{-28}-164q^{-30}+104q^{-32}-36q^{-34}-21q^{-36}+51q^{-38}-59q^{-40}+45q^{-42}-26q^{-44}+8q^{-46}+2q^{-48}-9q^{-50}+7q^{-52}-6q^{-54}+4q^{-56}-q^{-58}+q^{-60}$

Further Quantum Invariants

K11a231 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["K11a231"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 99, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+2q^{3}-5q^{2}+10q-12+16q^{-1}-16q^{-2}+14q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+10z^{4}+6a^{4}z^{2}-19a^{2}z^{2}-4z^{2}a^{-2}+18z^{2}+3a^{4}-10a^{2}-4a^{-2}+12$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+6az^{9}+2z^{9}a^{-1}+8a^{4}z^{8}+10a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+9a^{5}z^{7}+4a^{3}z^{7}-9az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-12a^{4}z^{6}-31a^{2}z^{6}-8z^{6}a^{-2}-21z^{6}+3a^{7}z^{5}-16a^{5}z^{5}-29a^{3}z^{5}-14az^{5}-9z^{5}a^{-1}-5z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+6a^{4}z^{4}+30a^{2}z^{4}+11z^{4}a^{-2}+28z^{4}-2a^{7}z^{3}+16a^{5}z^{3}+35a^{3}z^{3}+29az^{3}+20z^{3}a^{-1}+8z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-2a^{4}z^{2}-22a^{2}z^{2}-8z^{2}a^{-2}-24z^{2}-6a^{5}z-16a^{3}z-17az-11za^{-1}-4za^{-3}+3a^{4}+10a^{2}+4a^{-2}+12$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a108, K11a139,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a57,}

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["K11a231"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+2q^{3}-5q^{2}+10q-12+16q^{-1}-16q^{-2}+14q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{K11a57, K11a108, K11a139,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a57,}

Vassiliev invariants

V2 and V3:

(1, 3)

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 $4$ $24$ $8$ ${\frac {14}{3}}$ ${\frac {58}{3}}$ $96$ $80$ $-64$ $24$ ${\frac {32}{3}}$ $288$ ${\frac {56}{3}}$ ${\frac {232}{3}}$ ${\frac {19231}{30}}$ ${\frac {1258}{15}}$ ${\frac {13622}{45}}$ ${\frac {929}{18}}$ ${\frac {511}{30}}$

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 K11a231. 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                     1   1 5                   4 1   -3 3                 6 1     5 1               6 4       -2 -1             10 6         4 -3           7 7           0 -5         7 9             -2 -7       5 7               2 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a230 K11a232