K11a108

K11a107

K11a109

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a108 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
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:K11a108/ThurstonBennequinNumber Hyperbolic Volume 14.2368 A-Polynomial See Data:K11a108/A-polynomial

[edit Notes for K11a108'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 K11a108'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)	$\{1\}$
Determinant and Signature	{ 99, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+10q-13+16q^{-1}-15q^{-2}+14q^{-3}-11q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-14a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+a^{4}-4a^{2}-2a^{-2}+6$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+7a^{5}z^{7}+a^{3}z^{7}-14az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-11a^{4}z^{6}-38a^{2}z^{6}-12z^{6}a^{-2}-34z^{6}+3a^{7}z^{5}-9a^{5}z^{5}-17a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+9a^{4}z^{4}+41a^{2}z^{4}+15z^{4}a^{-2}+42z^{4}-3a^{7}z^{3}+5a^{5}z^{3}+15a^{3}z^{3}+14az^{3}+12z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-5a^{4}z^{2}-23a^{2}z^{2}-7z^{2}a^{-2}-23z^{2}+a^{7}z+a^{5}z-3a^{3}z-6az-5za^{-1}-2za^{-3}+a^{4}+4a^{2}+2a^{-2}+6$
The A2 invariant	$q^{20}-q^{18}+2q^{16}-2q^{14}-2q^{12}+q^{10}-3q^{8}+4q^{6}-q^{4}+2q^{2}+2-q^{-2}+3q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+5q^{106}-4q^{104}-2q^{102}+10q^{100}-18q^{98}+26q^{96}-30q^{94}+26q^{92}-11q^{90}-11q^{88}+42q^{86}-67q^{84}+82q^{82}-85q^{80}+58q^{78}-10q^{76}-54q^{74}+123q^{72}-159q^{70}+166q^{68}-121q^{66}+33q^{64}+71q^{62}-169q^{60}+217q^{58}-192q^{56}+95q^{54}+28q^{52}-133q^{50}+183q^{48}-144q^{46}+33q^{44}+91q^{42}-185q^{40}+182q^{38}-89q^{36}-77q^{34}+236q^{32}-310q^{30}+275q^{28}-132q^{26}-74q^{24}+261q^{22}-364q^{20}+344q^{18}-214q^{16}+19q^{14}+174q^{12}-275q^{10}+279q^{8}-174q^{6}+20q^{4}+124q^{2}-200+173q^{-2}-59q^{-4}-86q^{-6}+209q^{-8}-233q^{-10}+159q^{-12}-12q^{-14}-145q^{-16}+257q^{-18}-275q^{-20}+196q^{-22}-61q^{-24}-85q^{-26}+184q^{-28}-207q^{-30}+165q^{-32}-82q^{-34}-3q^{-36}+61q^{-38}-88q^{-40}+76q^{-42}-48q^{-44}+18q^{-46}+4q^{-48}-15q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a108 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["K11a108"];

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 99, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+3q^{3}-6q^{2}+10q-13+16q^{-1}-15q^{-2}+14q^{-3}-11q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+7a^{5}z^{7}+a^{3}z^{7}-14az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-11a^{4}z^{6}-38a^{2}z^{6}-12z^{6}a^{-2}-34z^{6}+3a^{7}z^{5}-9a^{5}z^{5}-17a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-4a^{6}z^{4}+9a^{4}z^{4}+41a^{2}z^{4}+15z^{4}a^{-2}+42z^{4}-3a^{7}z^{3}+5a^{5}z^{3}+15a^{3}z^{3}+14az^{3}+12z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-5a^{4}z^{2}-23a^{2}z^{2}-7z^{2}a^{-2}-23z^{2}+a^{7}z+a^{5}z-3a^{3}z-6az-5za^{-1}-2za^{-3}+a^{4}+4a^{2}+2a^{-2}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a139, K11a231,}

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["K11a108"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+3q^{3}-6q^{2}+10q-13+16q^{-1}-15q^{-2}+14q^{-3}-11q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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, K11a139, K11a231,}

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, 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 $4$ $8$ $8$ ${\frac {62}{3}}$ ${\frac {58}{3}}$ $32$ $-{\frac {112}{3}}$ $-{\frac {160}{3}}$ $-24$ ${\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {232}{3}}$ ${\frac {8911}{30}}$ ${\frac {446}{5}}$ ${\frac {7142}{45}}$ ${\frac {497}{18}}$ $-{\frac {209}{30}}$

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 K11a108. 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                     2   2 5                   4 1   -3 3                 6 2     4 1               7 4       -3 -1             9 6         3 -3           7 8           1 -5         7 8             -1 -7       4 7               3 -9     2 7                 -5 -11   1 4                   3 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a107 K11a109