K11a194

K11a193

K11a195

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a194 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a194/ThurstonBennequinNumber Hyperbolic Volume 13.7968 A-Polynomial See Data:K11a194/A-polynomial

[edit Notes for K11a194's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a194's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, -4 }
Jones polynomial	$-q+3-5q^{-1}+10q^{-2}-12q^{-3}+14q^{-4}-15q^{-5}+13q^{-6}-10q^{-7}+6q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-13z^{2}a^{6}-6a^{6}+z^{8}a^{4}+6z^{6}a^{4}+14z^{4}a^{4}+15z^{2}a^{4}+5a^{4}-z^{6}a^{2}-4z^{4}a^{2}-4z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-3z^{3}a^{11}+5z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}+7z^{7}a^{9}-12z^{5}a^{9}+11z^{3}a^{9}-4za^{9}+7z^{8}a^{8}-14z^{6}a^{8}+13z^{4}a^{8}-5z^{2}a^{8}+2a^{8}+4z^{9}a^{7}-z^{7}a^{7}-17z^{5}a^{7}+23z^{3}a^{7}-8za^{7}+z^{10}a^{6}+10z^{8}a^{6}-39z^{6}a^{6}+46z^{4}a^{6}-25z^{2}a^{6}+6a^{6}+7z^{9}a^{5}-18z^{7}a^{5}+7z^{5}a^{5}+5z^{3}a^{5}-3za^{5}+z^{10}a^{4}+6z^{8}a^{4}-33z^{6}a^{4}+45z^{4}a^{4}-27z^{2}a^{4}+5a^{4}+3z^{9}a^{3}-9z^{7}a^{3}+5z^{5}a^{3}+za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+18z^{4}a^{2}-9z^{2}a^{2}+z^{7}a-4z^{5}a+4z^{3}a$
The A2 invariant	$q^{30}+q^{24}-3q^{22}+q^{20}-2q^{18}-q^{16}+q^{14}-3q^{12}+4q^{10}+3q^{6}+2q^{4}-q^{2}+1-q^{-2}$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+5q^{154}-3q^{152}-2q^{150}+10q^{148}-17q^{146}+24q^{144}-27q^{142}+19q^{140}-7q^{138}-13q^{136}+36q^{134}-55q^{132}+68q^{130}-65q^{128}+41q^{126}+q^{124}-47q^{122}+96q^{120}-124q^{118}+125q^{116}-90q^{114}+17q^{112}+68q^{110}-135q^{108}+168q^{106}-138q^{104}+66q^{102}+28q^{100}-112q^{98}+143q^{96}-110q^{94}+17q^{92}+82q^{90}-147q^{88}+138q^{86}-54q^{84}-71q^{82}+185q^{80}-241q^{78}+202q^{76}-92q^{74}-74q^{72}+215q^{70}-286q^{68}+263q^{66}-150q^{64}-5q^{62}+143q^{60}-221q^{58}+215q^{56}-136q^{54}+6q^{52}+107q^{50}-157q^{48}+135q^{46}-36q^{44}-76q^{42}+165q^{40}-176q^{38}+112q^{36}-q^{34}-123q^{32}+211q^{30}-210q^{28}+146q^{26}-34q^{24}-75q^{22}+150q^{20}-163q^{18}+126q^{16}-62q^{14}-6q^{12}+53q^{10}-70q^{8}+61q^{6}-37q^{4}+16q^{2}+3-13q^{-2}+11q^{-4}-10q^{-6}+5q^{-8}-2q^{-10}+q^{-12}$

Further Quantum Invariants

K11a194 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["K11a194"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

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]=

{ 93, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q+3-5q^{-1}+10q^{-2}-12q^{-3}+14q^{-4}-15q^{-5}+13q^{-6}-10q^{-7}+6q^{-8}-3q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-13z^{2}a^{6}-6a^{6}+z^{8}a^{4}+6z^{6}a^{4}+14z^{4}a^{4}+15z^{2}a^{4}+5a^{4}-z^{6}a^{2}-4z^{4}a^{2}-4z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-3z^{3}a^{11}+5z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}+7z^{7}a^{9}-12z^{5}a^{9}+11z^{3}a^{9}-4za^{9}+7z^{8}a^{8}-14z^{6}a^{8}+13z^{4}a^{8}-5z^{2}a^{8}+2a^{8}+4z^{9}a^{7}-z^{7}a^{7}-17z^{5}a^{7}+23z^{3}a^{7}-8za^{7}+z^{10}a^{6}+10z^{8}a^{6}-39z^{6}a^{6}+46z^{4}a^{6}-25z^{2}a^{6}+6a^{6}+7z^{9}a^{5}-18z^{7}a^{5}+7z^{5}a^{5}+5z^{3}a^{5}-3za^{5}+z^{10}a^{4}+6z^{8}a^{4}-33z^{6}a^{4}+45z^{4}a^{4}-27z^{2}a^{4}+5a^{4}+3z^{9}a^{3}-9z^{7}a^{3}+5z^{5}a^{3}+za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+18z^{4}a^{2}-9z^{2}a^{2}+z^{7}a-4z^{5}a+4z^{3}a$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a346,}

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["K11a194"];

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]=

{ $t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$, $-q+3-5q^{-1}+10q^{-2}-12q^{-3}+14q^{-4}-15q^{-5}+13q^{-6}-10q^{-7}+6q^{-8}-3q^{-9}+q^{-10}$ }

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]=

{K11a106, K11a346,}

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 $4$ $8$ $8$ $-{\frac {226}{3}}$ $-{\frac {38}{3}}$ $32$ ${\frac {944}{3}}$ $-{\frac {64}{3}}$ $104$ ${\frac {32}{3}}$ $32$ $-{\frac {904}{3}}$ $-{\frac {152}{3}}$ $-{\frac {26129}{30}}$ ${\frac {5738}{15}}$ $-{\frac {24058}{45}}$ $-{\frac {2191}{18}}$ $-{\frac {2129}{30}}$

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 $s=$-4 is the signature of K11a194. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     2   2 -1                   3 1   -2 -3                 7 2     5 -5               6 4       -2 -7             8 6         2 -9           7 6           -1 -11         6 8             -2 -13       4 7               3 -15     2 6                 -4 -17   1 4                   3 -19   2                     -2 -21 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a193 K11a195