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

K11a84 K11a86

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 2 22 20 18 8 14 6

Alexander Polynomial:

2t-3 - 10t-2 + 25t-1 - 33 + 25t - 10t2 + 2t3

Conway Polynomial:

1 + 3z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{107, 2}

Jones Polynomial:

- q-2 + 3q-1 - 6 + 11q - 14q2 + 17q3 - 17q4 + 15q5 - 11q6 + 7q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 1 + 4q2 - 2q4 + 3q6 + 2q8 - q10 + 2q12 - 3q14 + 2q16 - 2q20 + 2q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - a-6 - 3a-6z2 - 2a-6z4 + 2a-4z2 + 2a-4z4 + a-4z6 + 3a-2 + 5a-2z2 + 3a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + 6a-9z3 - 11a-9z5 + 4a-9z7 - 3a-8z2 + 13a-8z4 - 17a-8z6 + 6a-8z8 - 3a-7z + 12a-7z3 - 10a-7z5 - 4a-7z7 + 4a-7z9 + a-6 - 11a-6z2 + 33a-6z4 - 33a-6z6 + 10a-6z8 + a-6z10 - 3a-5z + 6a-5z3 + 2a-5z5 - 11a-5z7 + 7a-5z9 - 5a-4z2 + 17a-4z4 - 19a-4z6 + 8a-4z8 + a-4z10 + a-3z - 4a-3z5 + a-3z7 + 3a-3z9 - 3a-2 + 7a-2z2 - 7a-2z4 - a-2z6 + 4a-2z8 + 2a-1z - 2a-1z3 - 4a-1z5 + 4a-1z7 - 1 + 4z2 - 6z4 + 3z6 + az - 2az3 + az5

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

{3, 4}

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 1185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   4 1   j = 13                 7 3     j = 11               8 4       j = 9             9 7         j = 7           8 8           j = 5         6 9             j = 3       5 8               j = 1     2 7                 j = -1   1 4                   j = -3   2                     j = -5 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, 85]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 85]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 14, 21, 13], X[18, 16, 19, 15], > X[8, 18, 9, 17], X[14, 20, 15, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 85]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 85]]
Out[5]=	DTCode[4, 10, 12, 16, 2, 22, 20, 18, 8, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 85]][t]
Out[6]=	2 10 25 2 3 -33 + -- - -- + -- + 25 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 85]][z]
Out[7]=	2 4 6 1 + 3 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 85]}
In[9]:=	{KnotDet[Knot[11, Alternating, 85]], KnotSignature[Knot[11, Alternating, 85]]}
Out[9]=	{107, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 85]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 9 -6 - q + - + 11 q - 14 q + 17 q - 17 q + 15 q - 11 q + 7 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 85]}
In[12]:=	A2Invariant[Knot[11, Alternating, 85]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 14 -1 - q + q - q + 4 q - 2 q + 3 q + 2 q - q + 2 q - 3 q + 16 20 22 24 26 28 > 2 q - 2 q + 2 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 85]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -6 3 2 z 3 z 2 z 5 z 4 2 z 2 z 3 z z -1 - a + -- - 2 z + -- - ---- + ---- + ---- - z - ---- + ---- + ---- + -- + 2 8 6 4 2 6 4 2 4 a a a a a a a a a 6 z > -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 85]][a, z]
Out[14]=	2 2 2 -6 3 3 z 3 z z 2 z 2 3 z 11 z 5 z -1 + a - -- - --- - --- + -- + --- + a z + 4 z - ---- - ----- - ---- + 2 7 5 3 a 8 6 4 a a a a a a a 2 3 3 3 3 4 4 4 7 z 6 z 12 z 6 z 2 z 3 4 2 z 13 z 33 z > ---- + ---- + ----- + ---- - ---- - 2 a z - 6 z - ---- + ----- + ----- + 2 9 7 5 a 10 8 6 a a a a a a a 4 4 5 5 5 5 5 6 17 z 7 z 11 z 10 z 2 z 4 z 4 z 5 6 z > ----- - ---- - ----- - ----- + ---- - ---- - ---- + a z + 3 z + --- - 4 2 9 7 5 3 a 10 a a a a a a a 6 6 6 6 7 7 7 7 7 8 17 z 33 z 19 z z 4 z 4 z 11 z z 4 z 6 z > ----- - ----- - ----- - -- + ---- - ---- - ----- + -- + ---- + ---- + 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 10 z 8 z 4 z 4 z 7 z 3 z z z > ----- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 85]], Vassiliev[3][Knot[11, Alternating, 85]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, Alternating, 85]][q, t]
Out[16]=	3 1 2 1 4 2 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 9 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 8 q t + 9 q t + 7 q t + 8 q t + 4 q t + 7 q t + 13 6 15 6 15 7 17 7 19 8 > 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a85

K11a84 K11a86