K11a88

K11a87

K11a89

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a88 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a88/ThurstonBennequinNumber Hyperbolic Volume 14.1444 A-Polynomial See Data:K11a88/A-polynomial

[edit Notes for K11a88's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a88's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-20t+25-20t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 101, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+11q^{2}-14q+16-16q^{-1}+14q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}+3a^{4}z^{2}-13a^{2}z^{2}-5z^{2}a^{-2}+14z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+7a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-16az^{7}-5z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-10a^{4}z^{6}-25a^{2}z^{6}-14z^{6}a^{-2}+3z^{6}a^{-4}-31z^{6}-9a^{5}z^{5}-4a^{3}z^{5}+18az^{5}+2z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+7a^{4}z^{4}+33a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+45z^{4}+7a^{5}z^{3}+2a^{3}z^{3}-10az^{3}+3z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-4a^{4}z^{2}-23a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}-26z^{2}-a^{5}z+az-za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	$q^{18}+q^{12}-3q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-3+4q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-2q^{92}+5q^{90}-9q^{88}+10q^{86}-10q^{84}+2q^{82}+15q^{80}-34q^{78}+55q^{76}-63q^{74}+49q^{72}-15q^{70}-43q^{68}+110q^{66}-157q^{64}+169q^{62}-122q^{60}+26q^{58}+100q^{56}-212q^{54}+273q^{52}-249q^{50}+137q^{48}+23q^{46}-180q^{44}+267q^{42}-248q^{40}+141q^{38}+20q^{36}-159q^{34}+209q^{32}-161q^{30}+8q^{28}+159q^{26}-270q^{24}+270q^{22}-147q^{20}-56q^{18}+264q^{16}-395q^{14}+397q^{12}-269q^{10}+41q^{8}+194q^{6}-357q^{4}+399q^{2}-297+117q^{-2}+85q^{-4}-224q^{-6}+254q^{-8}-171q^{-10}+19q^{-12}+132q^{-14}-205q^{-16}+178q^{-18}-50q^{-20}-109q^{-22}+236q^{-24}-274q^{-26}+216q^{-28}-84q^{-30}-82q^{-32}+206q^{-34}-259q^{-36}+232q^{-38}-136q^{-40}+24q^{-42}+70q^{-44}-129q^{-46}+139q^{-48}-114q^{-50}+67q^{-52}-16q^{-54}-22q^{-56}+41q^{-58}-45q^{-60}+37q^{-62}-23q^{-64}+11q^{-66}+q^{-68}-7q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

K11a88 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["K11a88"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-20t+25-20t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 101, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+11q^{2}-14q+16-16q^{-1}+14q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}+3a^{4}z^{2}-13a^{2}z^{2}-5z^{2}a^{-2}+14z^{2}+2a^{4}-5a^{2}-a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+7a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-16az^{7}-5z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-10a^{4}z^{6}-25a^{2}z^{6}-14z^{6}a^{-2}+3z^{6}a^{-4}-31z^{6}-9a^{5}z^{5}-4a^{3}z^{5}+18az^{5}+2z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+7a^{4}z^{4}+33a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+45z^{4}+7a^{5}z^{3}+2a^{3}z^{3}-10az^{3}+3z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-4a^{4}z^{2}-23a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}-26z^{2}-a^{5}z+az-za^{-1}-za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a84,}

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["K11a88"];

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]=

{ $t^{4}-5t^{3}+12t^{2}-20t+25-20t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$, $-q^{5}+3q^{4}-6q^{3}+11q^{2}-14q+16-16q^{-1}+14q^{-2}-10q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ }

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]=

{K11a84,}

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 $-4$ $16$ $8$ $-{\frac {62}{3}}$ $-{\frac {34}{3}}$ $-64$ $-{\frac {320}{3}}$ $-{\frac {320}{3}}$ $48$ $-{\frac {32}{3}}$ $128$ ${\frac {248}{3}}$ ${\frac {136}{3}}$ ${\frac {11729}{30}}$ $-{\frac {1778}{15}}$ ${\frac {18658}{45}}$ $-{\frac {1265}{18}}$ ${\frac {2609}{30}}$

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=$0 is the signature of K11a88. 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 11                       1 -1 9                     2   2 7                   4 1   -3 5                 7 2     5 3               7 4       -3 1             9 7         2 -1           8 8           0 -3         6 8             -2 -5       4 8               4 -7     2 6                 -4 -9   1 4                   3 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a87 K11a89