K11a158

K11a157

K11a159

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a158 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a158/ThurstonBennequinNumber Hyperbolic Volume 16.235 A-Polynomial See Data:K11a158/A-polynomial

[edit Notes for K11a158'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 K11a158's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-4z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$-q^{4}+4q^{3}-8q^{2}+13q-16+19q^{-1}-19q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+6a^{4}z^{2}-17a^{2}z^{2}-2z^{2}a^{-2}+11z^{2}+3a^{4}-7a^{2}-a^{-2}+6$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+12az^{9}+5z^{9}a^{-1}+10a^{4}z^{8}+14a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+9a^{5}z^{7}-7a^{3}z^{7}-30az^{7}-13z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-19a^{4}z^{6}-57a^{2}z^{6}-14z^{6}a^{-2}-46z^{6}+3a^{7}z^{5}-14a^{5}z^{5}-12a^{3}z^{5}+10az^{5}+2z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+17a^{4}z^{4}+65a^{2}z^{4}+15z^{4}a^{-2}+56z^{4}-2a^{7}z^{3}+12a^{5}z^{3}+18a^{3}z^{3}+10az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-9a^{4}z^{2}-34a^{2}z^{2}-6z^{2}a^{-2}-27z^{2}-4a^{5}z-7a^{3}z-5az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+a^{-2}+6$
The A2 invariant	$q^{20}-q^{18}+3q^{16}-q^{14}-q^{12}+2q^{10}-5q^{8}+2q^{6}-3q^{4}+q^{2}+3-q^{-2}+4q^{-4}-q^{-6}+q^{-10}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+6q^{106}-5q^{104}+9q^{100}-19q^{98}+29q^{96}-36q^{94}+31q^{92}-18q^{90}-6q^{88}+41q^{86}-71q^{84}+95q^{82}-101q^{80}+80q^{78}-38q^{76}-28q^{74}+110q^{72}-184q^{70}+232q^{68}-221q^{66}+138q^{64}+15q^{62}-189q^{60}+341q^{58}-389q^{56}+299q^{54}-92q^{52}-167q^{50}+361q^{48}-401q^{46}+268q^{44}-5q^{42}-260q^{40}+396q^{38}-338q^{36}+88q^{34}+233q^{32}-489q^{30}+543q^{28}-376q^{26}+39q^{24}+334q^{22}-601q^{20}+659q^{18}-500q^{16}+167q^{14}+201q^{12}-487q^{10}+590q^{8}-473q^{6}+202q^{4}+126q^{2}-368+438q^{-2}-298q^{-4}+21q^{-6}+276q^{-8}-444q^{-10}+409q^{-12}-173q^{-14}-149q^{-16}+430q^{-18}-533q^{-20}+435q^{-22}-184q^{-24}-118q^{-26}+340q^{-28}-418q^{-30}+346q^{-32}-175q^{-34}-7q^{-36}+131q^{-38}-178q^{-40}+153q^{-42}-91q^{-44}+30q^{-46}+14q^{-48}-32q^{-50}+28q^{-52}-19q^{-54}+9q^{-56}-3q^{-58}+q^{-60}$

Further Quantum Invariants

K11a158 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["K11a158"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 119, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-8q^{2}+13q-16+19q^{-1}-19q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+6a^{4}z^{2}-17a^{2}z^{2}-2z^{2}a^{-2}+11z^{2}+3a^{4}-7a^{2}-a^{-2}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+12az^{9}+5z^{9}a^{-1}+10a^{4}z^{8}+14a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+9a^{5}z^{7}-7a^{3}z^{7}-30az^{7}-13z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-19a^{4}z^{6}-57a^{2}z^{6}-14z^{6}a^{-2}-46z^{6}+3a^{7}z^{5}-14a^{5}z^{5}-12a^{3}z^{5}+10az^{5}+2z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+17a^{4}z^{4}+65a^{2}z^{4}+15z^{4}a^{-2}+56z^{4}-2a^{7}z^{3}+12a^{5}z^{3}+18a^{3}z^{3}+10az^{3}+9z^{3}a^{-1}+3z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-9a^{4}z^{2}-34a^{2}z^{2}-6z^{2}a^{-2}-27z^{2}-4a^{5}z-7a^{3}z-5az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+a^{-2}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a34,}

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["K11a158"];

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]=

{ $-t^{4}+5t^{3}-14t^{2}+25t-29+25t^{-1}-14t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+4q^{3}-8q^{2}+13q-16+19q^{-1}-19q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{K11a34,}

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:

(-2, 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 $-8$ $24$ $32$ ${\frac {164}{3}}$ ${\frac {148}{3}}$ $-192$ $-432$ $-128$ $-136$ $-{\frac {256}{3}}$ $288$ $-{\frac {1312}{3}}$ $-{\frac {1184}{3}}$ ${\frac {6449}{15}}$ ${\frac {6884}{15}}$ $-{\frac {25684}{45}}$ ${\frac {2047}{9}}$ $-{\frac {3631}{15}}$

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 K11a158. 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                     3   3 5                   5 1   -4 3                 8 3     5 1               8 5       -3 -1             11 8         3 -3           9 9           0 -5         7 10             -3 -7       5 9               4 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a157 K11a159