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

K11a68 K11a70

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X12,18,13,17 X10,20,11,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 14 2 18 20 22 12 10 6

Alexander Polynomial:

- 2t-3 + 13t-2 - 33t-1 + 45 - 33t + 13t2 - 2t3

Conway Polynomial:

1 + z2 + z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{141, 0}

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 15q-2 - 20q-1 + 23 - 22q + 20q2 - 14q3 + 8q4 - 4q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 4q-10 + 2q-8 - 3q-4 + 5q-2 - 2 + 4q2 - q6 + 4q8 - 5q10 + q12 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 6a-5z3 - 10a-5z5 + 4a-5z7 - 4a-4z2 + 15a-4z4 - 18a-4z6 + 7a-4z8 - a-3z + 8a-3z3 - 4a-3z5 - 8a-3z7 + 6a-3z9 + a-2 - 13a-2z2 + 34a-2z4 - 37a-2z6 + 12a-2z8 + 2a-2z10 - 2a-1z + 5a-1z3 + a-1z5 - 17a-1z7 + 12a-1z9 + 3 - 12z2 + 24z4 - 32z6 + 14z8 + 2z10 - 2az + 11az3 - 18az5 + 3az7 + 6az9 + a2 - a2z2 + 2a2z4 - 10a2z6 + 9a2z8 - a3z + 7a3z3 - 12a3z5 + 8a3z7 + 2a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

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

{1, 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 1169. 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                 9 3     j = 5               11 5       j = 3             11 9         j = 1           12 11           j = -1         9 12             j = -3       6 11               j = -5     3 9                 j = -7   1 6                   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, 69]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 69]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[12, 18, 13, 17], X[10, 20, 11, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 69]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -4, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 69]]
Out[5]=	DTCode[4, 8, 16, 14, 2, 18, 20, 22, 12, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 69]][t]
Out[6]=	2 13 33 2 3 45 - -- + -- - -- - 33 t + 13 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 69]][z]
Out[7]=	2 4 6 1 + z + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 69]}
In[9]:=	{KnotDet[Knot[11, Alternating, 69]], KnotSignature[Knot[11, Alternating, 69]]}
Out[9]=	{141, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 69]][q]
Out[10]=	-5 4 9 15 20 2 3 4 5 6 23 - q + -- - -- + -- - -- - 22 q + 20 q - 14 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, 69]}
In[12]:=	A2Invariant[Knot[11, Alternating, 69]][q]
Out[12]=	-16 -14 2 4 2 3 5 2 6 8 10 12 -2 - q + q + --- - --- + -- - -- + -- + 4 q - q + 4 q - 5 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, 69]][a, z]
Out[13]=	2 2 4 4 6 -2 2 2 z 3 z 2 2 4 2 z 2 z 2 4 6 z 3 - a - a + 3 z + -- - ---- + a z - a z + -- - ---- + 2 a z - z - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 69]][a, z]
Out[14]=	2 2 -2 2 z 2 z 3 2 4 z 13 z 2 2 3 + a + a - -- - --- - 2 a z - a z - 12 z - ---- - ----- - a z + 3 a 4 2 a a a 3 3 3 4 4 2 6 z 8 z 5 z 3 3 3 5 3 4 2 z > 2 a z + ---- + ---- + ---- + 11 a z + 7 a z - a z + 24 z - ---- + 5 3 a 6 a a a 4 4 5 5 5 15 z 34 z 2 4 4 4 10 z 4 z z 5 > ----- + ----- + 2 a z - 5 a z - ----- - ---- + -- - 18 a z - 4 2 5 3 a a a a a 6 6 6 7 3 5 5 5 6 z 18 z 37 z 2 6 4 6 4 z > 12 a z + a z - 32 z + -- - ----- - ----- - 10 a z + 4 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 9 8 z 17 z 7 3 7 8 7 z 12 z 2 8 6 z > ---- - ----- + 3 a z + 8 a z + 14 z + ---- + ----- + 9 a z + ---- + 3 a 4 2 3 a a a a 9 10 12 z 9 10 2 z > ----- + 6 a z + 2 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 69]], Vassiliev[3][Knot[11, Alternating, 69]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 69]][q, t]
Out[16]=	12 1 3 1 6 3 9 6 11 -- + 12 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 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 11 q t + 9 q t + 11 q t + 5 q t + 9 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 K11a69

K11a68 K11a70