© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a257 K11a259

Knotscape

This page is passe. Go here instead!    The Knot K11a258 Visit K11a258's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X20,11,21,12 X18,13,19,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22

Gauss Code:

{1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, -6, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 8 14 2 16 20 18 4 22 12 10

Alexander Polynomial:

2t-3 - 9t-2 + 17t-1 - 19 + 17t - 9t2 + 2t3

Conway Polynomial:

1 - z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11n150, ...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

q-5 - 2q-4 + 4q-3 - 7q-2 + 9q-1 - 11 + 12q - 10q2 + 9q3 - 6q4 + 3q5 - q6

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-16 + q-14 + q-10 - 2q-8 - 2q-6 - 2q-2 + 2 + q2 + q4 + 3q6 - q8 + 2q10 - q12 - q14 + q16 - q18

HOMFLY-PT Polynomial:

- a-4 - 2a-4z2 - a-4z4 + 2a-2 + 3a-2z2 + 3a-2z4 + a-2z6 + 2 + 3z2 + 3z4 + z6 - 4a2 - 6a2z2 - 2a2z4 + 2a4 + a4z2

Kauffman Polynomial:

a-7z3 + 3a-6z4 + a-5z - 4a-5z3 + 6a-5z5 - a-4 + 7a-4z2 - 14a-4z4 + 9a-4z6 + a-3z + 4a-3z3 - 16a-3z5 + 9a-3z7 - 2a-2 + 4a-2z2 - a-2z4 - 13a-2z6 + 7a-2z8 + 2a-1z + 4a-1z3 - 7a-1z5 - 6a-1z7 + 4a-1z9 + 2 - 19z2 + 43z4 - 34z6 + 6z8 + z10 + 6az - 19az3 + 35az5 - 26az7 + 6az9 + 4a2 - 25a2z2 + 39a2z4 - 18a2z6 + a2z10 + 4a3z - 14a3z3 + 20a3z5 - 11a3z7 + 2a3z9 + 2a4 - 9a4z2 + 12a4z4 - 6a4z6 + a4z8

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 3}

Khovanov Homology: (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 11258. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                       1 j = 11                     2   j = 9                   4 1   j = 7                 5 2     j = 5               5 4       j = 3             7 5         j = 1           5 6           j = -1         4 6             j = -3       3 5               j = -5     1 4                 j = -7   1 3                   j = -9   1                     j = -11 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 258]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 258]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[20, 11, 21, 12], X[18, 13, 19, 14], X[4, 15, 5, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 258]]
Out[4]=	GaussCode[1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 258]]
Out[5]=	DTCode[6, 8, 14, 2, 16, 20, 18, 4, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 258]][t]
Out[6]=	2 9 17 2 3 -19 + -- - -- + -- + 17 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 258]][z]
Out[7]=	2 4 6 1 - z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 258], Knot[11, NonAlternating, 150]}
In[9]:=	{KnotDet[Knot[11, Alternating, 258]], KnotSignature[Knot[11, Alternating, 258]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 258]][q]
Out[10]=	-5 2 4 7 9 2 3 4 5 6 -11 + q - -- + -- - -- + - + 12 q - 10 q + 9 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 258]}
In[12]:=	A2Invariant[Knot[11, Alternating, 258]][q]
Out[12]=	-16 -14 -10 2 2 2 2 4 6 8 10 12 2 + q + q + q - -- - -- - -- + q + q + 3 q - q + 2 q - q - 8 6 2 q q q 14 16 18 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 258]][a, z]
Out[13]=	2 2 4 -4 2 2 4 2 2 z 3 z 2 2 4 2 4 z 2 - a + -- - 4 a + 2 a + 3 z - ---- + ---- - 6 a z + a z + 3 z - -- + 2 4 2 4 a a a a 4 6 3 z 2 4 6 z > ---- - 2 a z + z + -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 258]][a, z]
Out[14]=	2 -4 2 2 4 z z 2 z 3 2 7 z 2 - a - -- + 4 a + 2 a + -- + -- + --- + 6 a z + 4 a z - 19 z + ---- + 2 5 3 a 4 a a a a 2 3 3 3 3 4 z 2 2 4 2 z 4 z 4 z 4 z 3 3 3 > ---- - 25 a z - 9 a z + -- - ---- + ---- + ---- - 19 a z - 14 a z + 2 7 5 3 a a a a a 4 4 4 5 5 5 4 3 z 14 z z 2 4 4 4 6 z 16 z 7 z > 43 z + ---- - ----- - -- + 39 a z + 12 a z + ---- - ----- - ---- + 6 4 2 5 3 a a a a a a 6 6 7 5 3 5 6 9 z 13 z 2 6 4 6 9 z > 35 a z + 20 a z - 34 z + ---- - ----- - 18 a z - 6 a z + ---- - 4 2 3 a a a 7 8 9 6 z 7 3 7 8 7 z 4 8 4 z 9 3 9 > ---- - 26 a z - 11 a z + 6 z + ---- + a z + ---- + 6 a z + 2 a z + a 2 a a 10 2 10 > z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 258]], Vassiliev[3][Knot[11, Alternating, 258]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 258]][q, t]
Out[16]=	3 1 1 1 3 1 4 3 5 6 q + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 4 6 5 q 3 5 5 2 7 2 7 3 > ---- + --- + --- + 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 2 q t t q t 9 3 9 4 11 4 13 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a258

K11a257 K11a259