K11a217

K11a216

K11a218

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a217 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a217/ThurstonBennequinNumber Hyperbolic Volume 17.2462 A-Polynomial See Data:K11a217/A-polynomial

[edit Notes for K11a217'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 K11a217's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+29t-35+29t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 139, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-15+20q^{-1}-22q^{-2}+23q^{-3}-19q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+12a^{4}z^{2}-9a^{2}z^{2}+3z^{2}-3a^{6}+6a^{4}-3a^{2}+1$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+8a^{6}z^{8}+14a^{4}z^{8}+13a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-a^{5}z^{7}-16a^{3}z^{7}-5az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-13a^{6}z^{6}-38a^{4}z^{6}-38a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-9a^{7}z^{5}-13a^{5}z^{5}-5a^{3}z^{5}-11az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+14a^{6}z^{4}+38a^{4}z^{4}+31a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-2a^{9}z^{3}+7a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+10az^{3}+6z^{3}a^{-1}+a^{8}z^{2}-9a^{6}z^{2}-20a^{4}z^{2}-14a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+a^{9}z-3a^{7}z-7a^{5}z-5a^{3}z-3az-za^{-1}+3a^{6}+6a^{4}+3a^{2}+1$
The A2 invariant	$-q^{24}-q^{20}-3q^{18}+4q^{16}-2q^{14}+3q^{12}+3q^{10}-2q^{8}+5q^{6}-5q^{4}+3q^{2}-1-2q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+10q^{120}-10q^{118}+4q^{116}+10q^{114}-29q^{112}+52q^{110}-73q^{108}+76q^{106}-58q^{104}+5q^{102}+83q^{100}-182q^{98}+271q^{96}-306q^{94}+242q^{92}-75q^{90}-188q^{88}+468q^{86}-660q^{84}+663q^{82}-438q^{80}+11q^{78}+468q^{76}-819q^{74}+889q^{72}-622q^{70}+115q^{68}+415q^{66}-748q^{64}+719q^{62}-337q^{60}-214q^{58}+689q^{56}-838q^{54}+581q^{52}-653q^{48}+1119q^{46}-1184q^{44}+816q^{42}-131q^{40}-614q^{38}+1165q^{36}-1312q^{34}+1024q^{32}-406q^{30}-307q^{28}+840q^{26}-1013q^{24}+772q^{22}-239q^{20}-347q^{18}+723q^{16}-730q^{14}+362q^{12}+203q^{10}-707q^{8}+918q^{6}-745q^{4}+257q^{2}+326-774q^{-2}+931q^{-4}-752q^{-6}+356q^{-8}+92q^{-10}-432q^{-12}+563q^{-14}-494q^{-16}+304q^{-18}-82q^{-20}-88q^{-22}+171q^{-24}-178q^{-26}+131q^{-28}-67q^{-30}+18q^{-32}+14q^{-34}-25q^{-36}+22q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a217 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["K11a217"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+29t-35+29t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 139, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-15+20q^{-1}-22q^{-2}+23q^{-3}-19q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+12a^{4}z^{2}-9a^{2}z^{2}+3z^{2}-3a^{6}+6a^{4}-3a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+8a^{6}z^{8}+14a^{4}z^{8}+13a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-a^{5}z^{7}-16a^{3}z^{7}-5az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-13a^{6}z^{6}-38a^{4}z^{6}-38a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-9a^{7}z^{5}-13a^{5}z^{5}-5a^{3}z^{5}-11az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+14a^{6}z^{4}+38a^{4}z^{4}+31a^{2}z^{4}-2z^{4}a^{-2}+9z^{4}-2a^{9}z^{3}+7a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+10az^{3}+6z^{3}a^{-1}+a^{8}z^{2}-9a^{6}z^{2}-20a^{4}z^{2}-14a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+a^{9}z-3a^{7}z-7a^{5}z-5a^{3}z-3az-za^{-1}+3a^{6}+6a^{4}+3a^{2}+1$

"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["K11a217"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+29t-35+29t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-15+20q^{-1}-22q^{-2}+23q^{-3}-19q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$ }

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:

(3, -6)

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 $12$ $-48$ $72$ $206$ $34$ $-576$ $-1056$ $-128$ $-208$ $288$ $1152$ $2472$ $408$ ${\frac {54591}{10}}$ $-{\frac {1022}{5}}$ ${\frac {36782}{15}}$ ${\frac {737}{6}}$ ${\frac {3231}{10}}$

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 K11a217. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 9 3     -6 -1               11 6       5 -3             12 10         -2 -5           11 10           1 -7         8 12             4 -9       6 11               -5 -11     2 8                 6 -13   1 6                   -5 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a216 K11a218