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

K11a94 K11a96

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X22,14,1,13 X20,16,21,15 X6,18,7,17 X16,20,17,19 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 18 2 8 22 20 6 16 14

Alexander Polynomial:

6t-2 - 18t-1 + 25 - 18t + 6t2

Conway Polynomial:

1 + 6z2 + 6z4

Other knots with the same Alexander/Conway Polynomial:

{1053, ...}

Determinant and Signature:

{73, 4}

Jones Polynomial:

q2 - 2q3 + 5q4 - 7q5 + 10q6 - 11q7 + 11q8 - 10q9 + 7q10 - 5q11 + 3q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - q8 + 2q10 + q12 - q14 + 3q16 + q22 - q24 + q26 - 2q28 - q30 + q32 - 2q34 + q36 + q38 - q40

HOMFLY-PT Polynomial:

- a-12z2 - a-10 + a-10z4 + 2a-8z2 + 2a-8z4 + a-6 + 3a-6z2 + 2a-6z4 + a-4 + 2a-4z2 + a-4z4

Kauffman Polynomial:

- a-15z + 4a-15z3 - 4a-15z5 + a-15z7 - 4a-14z2 + 15a-14z4 - 13a-14z6 + 3a-14z8 - a-13z + 9a-13z5 - 11a-13z7 + 3a-13z9 - 4a-12z2 + 18a-12z4 - 17a-12z6 + 2a-12z8 + a-12z10 - 3a-11z + a-11z3 + 10a-11z5 - 15a-11z7 + 5a-11z9 + a-10 - 5a-10z2 + 9a-10z4 - 10a-10z6 + 2a-10z8 + a-10z10 - a-9z + 5a-9z3 - 6a-9z5 + 2a-9z9 - a-8z2 + 2a-8z4 - 3a-8z6 + 3a-8z8 + 2a-7z - 2a-7z3 - a-7z5 + 3a-7z7 - a-6 + 2a-6z2 - 3a-6z4 + 3a-6z6 - 2a-5z3 + 2a-5z5 + a-4 - 2a-4z2 + a-4z4

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

{6, 15}

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=4 is the signature of 1195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 27                       1 j = 25                     2   j = 23                   3 1   j = 21                 4 2     j = 19               6 3       j = 17             5 4         j = 15           6 6           j = 13         4 5             j = 11       3 6               j = 9     2 4                 j = 7     3                   j = 5 1 2                     j = 3 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, 95]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 95]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 8, 19, 7], > X[2, 10, 3, 9], X[8, 12, 9, 11], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[6, 18, 7, 17], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 95]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 95]]
Out[5]=	DTCode[4, 10, 12, 18, 2, 8, 22, 20, 6, 16, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 95]][t]
Out[6]=	6 18 2 25 + -- - -- - 18 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 95]][z]
Out[7]=	2 4 1 + 6 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 53], Knot[11, Alternating, 95]}
In[9]:=	{KnotDet[Knot[11, Alternating, 95]], KnotSignature[Knot[11, Alternating, 95]]}
Out[9]=	{73, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 95]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 5 q - 7 q + 10 q - 11 q + 11 q - 10 q + 7 q - 5 q + 12 13 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 95]}
In[12]:=	A2Invariant[Knot[11, Alternating, 95]][q]
Out[12]=	6 8 10 12 14 16 22 24 26 28 30 32 q - q + 2 q + q - q + 3 q + q - q + q - 2 q - q + q - 34 36 38 40 > 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 95]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -10 -6 -4 z 2 z 3 z 2 z z 2 z 2 z z -a + a + a - --- + ---- + ---- + ---- + --- + ---- + ---- + -- 12 8 6 4 10 8 6 4 a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 95]][a, z]
Out[14]=	2 2 2 2 -10 -6 -4 z z 3 z z 2 z 4 z 4 z 5 z z a - a + a - --- - --- - --- - -- + --- - ---- - ---- - ---- - -- + 15 13 11 9 7 14 12 10 8 a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 2 z 2 z 4 z z 5 z 2 z 2 z 15 z 18 z 9 z > ---- - ---- + ---- + --- + ---- - ---- - ---- + ----- + ----- + ---- + 6 4 15 11 9 7 5 14 12 10 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 6 2 z 3 z z 4 z 9 z 10 z 6 z z 2 z 13 z 17 z > ---- - ---- + -- - ---- + ---- + ----- - ---- - -- + ---- - ----- - ----- - 8 6 4 15 13 11 9 7 5 14 12 a a a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 10 z 3 z 3 z z 11 z 15 z 3 z 3 z 2 z 2 z > ----- - ---- + ---- + --- - ----- - ----- + ---- + ---- + ---- + ---- + 10 8 6 15 13 11 7 14 12 10 a a a a a a a a a a 8 9 9 9 10 10 3 z 3 z 5 z 2 z z z > ---- + ---- + ---- + ---- + --- + --- 8 13 11 9 12 10 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 95]], Vassiliev[3][Knot[11, Alternating, 95]]}
Out[15]=	{6, 15}
In[16]:=	Kh[Knot[11, Alternating, 95]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 3 q t + 2 q t + 4 q t + 3 q t + 6 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 4 q t + 5 q t + 6 q t + 6 q t + 5 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 23 10 > 6 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 25 10 27 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a95

K11a94 K11a96