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

K11a90 K11a92

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 20 2 22 18 8 6 14

Alexander Polynomial:

- 2t-3 + 12t-2 - 30t-1 + 41 - 30t + 12t2 - 2t3

Conway Polynomial:

1 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{129, 0}

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 13q-2 - 18q-1 + 21 - 20q + 18q2 - 13q3 + 8q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 3q-10 + 2q-8 - 3q-4 + 4q-2 - 2 + 3q2 - q6 + 4q8 - 4q10 + q12 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - 2a-2z2 - 2a-2z4 - a-2z6 + 1 - z4 - z6 + 2a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - a-5z + 7a-5z3 - 10a-5z5 + 4a-5z7 - a-4z2 + 7a-4z4 - 13a-4z6 + 6a-4z8 - 4a-3z + 19a-3z3 - 22a-3z5 + 2a-3z7 + 4a-3z9 - 7a-2z2 + 25a-2z4 - 35a-2z6 + 14a-2z8 + a-2z10 - 6a-1z + 25a-1z3 - 26a-1z5 - a-1z7 + 8a-1z9 + 1 - 5z2 + 20z4 - 31z6 + 15z8 + z10 - 4az + 20az3 - 26az5 + 8az7 + 4az9 + 2a2z2 - 2a2z4 - 6a2z6 + 7a2z8 - a3z + 6a3z3 - 11a3z5 + 7a3z7 + 2a4z2 - 6a4z4 + 4a4z6 - a5z3 + a5z5

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

{0, 0}

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=0 is the signature of 1191. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   5 1   j = 7                 8 3     j = 5               10 5       j = 3             10 8         j = 1           11 10           j = -1         8 11             j = -3       5 10               j = -5     3 8                 j = -7   1 5                   j = -9   3                     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, 91]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 91]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[22, 14, 1, 13], X[18, 15, 19, 16], > X[8, 17, 9, 18], X[6, 20, 7, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 91]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -8, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 91]]
Out[5]=	DTCode[4, 10, 12, 16, 20, 2, 22, 18, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 91]][t]
Out[6]=	2 12 30 2 3 41 - -- + -- - -- - 30 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 91]][z]
Out[7]=	6 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 91]}
In[9]:=	{KnotDet[Knot[11, Alternating, 91]], KnotSignature[Knot[11, Alternating, 91]]}
Out[9]=	{129, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 91]][q]
Out[10]=	-5 4 8 13 18 2 3 4 5 6 21 - q + -- - -- + -- - -- - 20 q + 18 q - 13 q + 8 q - 4 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, 91]}
In[12]:=	A2Invariant[Knot[11, Alternating, 91]][q]
Out[12]=	-16 -14 2 3 2 3 4 2 6 8 10 12 -2 - q + q + --- - --- + -- - -- + -- + 3 q - q + 4 q - 4 q + q + 12 10 8 4 2 q q q q q 14 16 18 > q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 91]][a, z]
Out[13]=	2 2 4 4 6 z 2 z 2 2 4 2 4 z 2 z 2 4 6 z 1 + -- - ---- + 2 a z - a z - z + -- - ---- + 2 a z - z - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 91]][a, z]
Out[14]=	2 2 2 z 4 z 6 z 3 2 z z 7 z 2 2 4 2 1 - -- - --- - --- - 4 a z - a z - 5 z + -- - -- - ---- + 2 a z + 2 a z + 5 3 a 6 4 2 a a a a a 3 3 3 4 4 7 z 19 z 25 z 3 3 3 5 3 4 2 z 7 z > ---- + ----- + ----- + 20 a z + 6 a z - a z + 20 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 25 z 2 4 4 4 10 z 22 z 26 z 5 3 5 > ----- - 2 a z - 6 a z - ----- - ----- - ----- - 26 a z - 11 a z + 2 5 3 a a a a 6 6 6 7 7 7 5 5 6 z 13 z 35 z 2 6 4 6 4 z 2 z z > a z - 31 z + -- - ----- - ----- - 6 a z + 4 a z + ---- + ---- - -- + 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 6 z 14 z 2 8 4 z 8 z 9 > 8 a z + 7 a z + 15 z + ---- + ----- + 7 a z + ---- + ---- + 4 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 91]], Vassiliev[3][Knot[11, Alternating, 91]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 91]][q, t]
Out[16]=	11 1 3 1 5 3 8 5 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a91

K11a90 K11a92