K11a142

K11a141

K11a143

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a142 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a142/ThurstonBennequinNumber Hyperbolic Volume 11.4138 A-Polynomial See Data:K11a142/A-polynomial

[edit Notes for K11a142's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 6

[edit Notes for K11a142's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-8t^{2}+10t-11+10t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+2z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, -6 }
Jones polynomial	$q^{-1}-2q^{-2}+4q^{-3}-6q^{-4}+8q^{-5}-8q^{-6}+9q^{-7}-8q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+14z^{2}a^{8}+5a^{8}-z^{8}a^{6}-6z^{6}a^{6}-12z^{4}a^{6}-10z^{2}a^{6}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+7z^{2}a^{4}+2a^{4}$
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-2z^{3}a^{13}+za^{13}+4z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-11z^{5}a^{11}+7z^{3}a^{11}-2za^{11}+5z^{8}a^{10}-15z^{6}a^{10}+15z^{4}a^{10}-11z^{2}a^{10}+3a^{10}+3z^{9}a^{9}-6z^{7}a^{9}-6z^{5}a^{9}+10z^{3}a^{9}-3za^{9}+z^{10}a^{8}+3z^{8}a^{8}-26z^{6}a^{8}+38z^{4}a^{8}-21z^{2}a^{8}+5a^{8}+5z^{9}a^{7}-22z^{7}a^{7}+27z^{5}a^{7}-11z^{3}a^{7}+2za^{7}+z^{10}a^{6}-z^{8}a^{6}-13z^{6}a^{6}+28z^{4}a^{6}-16z^{2}a^{6}+3a^{6}+2z^{9}a^{5}-11z^{7}a^{5}+19z^{5}a^{5}-11z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+12z^{4}a^{4}-9z^{2}a^{4}+2a^{4}$
The A2 invariant	$-q^{36}-q^{34}-q^{30}+q^{28}-q^{26}+q^{24}+q^{22}+3q^{18}-q^{16}+q^{14}-q^{12}+q^{8}+q^{4}$
The G2 invariant	$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{188}-2q^{184}+4q^{182}-5q^{180}+6q^{178}-5q^{176}+2q^{174}+2q^{172}-5q^{170}+9q^{168}-11q^{166}+8q^{164}-7q^{162}+5q^{158}-10q^{156}+14q^{154}-13q^{152}+9q^{150}-4q^{148}-5q^{146}+7q^{144}-10q^{142}+12q^{140}-13q^{138}+14q^{136}-11q^{134}+3q^{132}+8q^{130}-19q^{128}+24q^{126}-25q^{124}+11q^{122}+4q^{120}-20q^{118}+24q^{116}-12q^{114}-8q^{112}+23q^{110}-29q^{108}+14q^{106}+11q^{104}-33q^{102}+46q^{100}-40q^{98}+26q^{96}+8q^{94}-30q^{92}+48q^{90}-46q^{88}+35q^{86}-12q^{84}-9q^{82}+27q^{80}-33q^{78}+30q^{76}-12q^{74}-10q^{72}+24q^{70}-29q^{68}+11q^{66}+13q^{64}-34q^{62}+41q^{60}-33q^{58}+6q^{56}+24q^{54}-44q^{52}+49q^{50}-36q^{48}+11q^{46}+14q^{44}-27q^{42}+30q^{40}-21q^{38}+12q^{36}+q^{34}-6q^{32}+7q^{30}-6q^{28}+4q^{26}-q^{24}+q^{22}$

Further Quantum Invariants

K11a142 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["K11a142"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-8t^{2}+10t-11+10t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}+2z^{4}+7z^{2}+1$

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]=

{ 59, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-1}-2q^{-2}+4q^{-3}-6q^{-4}+8q^{-5}-8q^{-6}+9q^{-7}-8q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+14z^{2}a^{8}+5a^{8}-z^{8}a^{6}-6z^{6}a^{6}-12z^{4}a^{6}-10z^{2}a^{6}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+7z^{2}a^{4}+2a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-2z^{3}a^{13}+za^{13}+4z^{6}a^{12}-5z^{4}a^{12}+2z^{2}a^{12}+5z^{7}a^{11}-11z^{5}a^{11}+7z^{3}a^{11}-2za^{11}+5z^{8}a^{10}-15z^{6}a^{10}+15z^{4}a^{10}-11z^{2}a^{10}+3a^{10}+3z^{9}a^{9}-6z^{7}a^{9}-6z^{5}a^{9}+10z^{3}a^{9}-3za^{9}+z^{10}a^{8}+3z^{8}a^{8}-26z^{6}a^{8}+38z^{4}a^{8}-21z^{2}a^{8}+5a^{8}+5z^{9}a^{7}-22z^{7}a^{7}+27z^{5}a^{7}-11z^{3}a^{7}+2za^{7}+z^{10}a^{6}-z^{8}a^{6}-13z^{6}a^{6}+28z^{4}a^{6}-16z^{2}a^{6}+3a^{6}+2z^{9}a^{5}-11z^{7}a^{5}+19z^{5}a^{5}-11z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+12z^{4}a^{4}-9z^{2}a^{4}+2a^{4}$

"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["K11a142"];

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]=

{ $-t^{4}+5t^{3}-8t^{2}+10t-11+10t^{-1}-8t^{-2}+5t^{-3}-t^{-4}$, $q^{-1}-2q^{-2}+4q^{-3}-6q^{-4}+8q^{-5}-8q^{-6}+9q^{-7}-8q^{-8}+6q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$ }

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:

(7, -20)

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 $28$ $-160$ $392$ ${\frac {3314}{3}}$ ${\frac {526}{3}}$ $-4480$ $-{\frac {25408}{3}}$ $-{\frac {4384}{3}}$ $-1216$ ${\frac {10976}{3}}$ $12800$ ${\frac {92792}{3}}$ ${\frac {14728}{3}}$ ${\frac {1979977}{30}}$ ${\frac {12206}{15}}$ ${\frac {1211474}{45}}$ ${\frac {10295}{18}}$ ${\frac {107017}{30}}$

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 $s=$-6 is the signature of K11a142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                       1 1 -3                     1   -1 -5                   3 1   2 -7                 4 2     -2 -9               4 2       2 -11             4 4         0 -13           5 4           1 -15         3 4             1 -17       3 5               -2 -19     1 3                 2 -21   1 3                   -2 -23   1                     1 -25 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a141 K11a143