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

K11a66 K11a68

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 12 2 18 22 20 10 6 14

Alexander Polynomial:

- 2t-3 + 12t-2 - 29t-1 + 39 - 29t + 12t2 - 2t3

Conway Polynomial:

1 + z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a104, K11a168, ...}

Determinant and Signature:

{125, 0}

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 12q-4 - 16q-3 + 20q-2 - 20q-1 + 18 - 14q + 9q2 - 4q3 + q4

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

{K11a317, ...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - 3q-16 + 2q-14 + q-12 - 2q-10 + 5q-8 - q-6 + 2q-4 - 3 + 3q2 - 4q4 + 2q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 - 2 - 3z2 - 2z4 - z6 + 2a2 + a2z2 - a2z4 - a2z6 + a4 + 3a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 - a-2 + 3a-2z2 - 8a-2z4 + 9a-2z6 - 2a-1z + 9a-1z3 - 18a-1z5 + 13a-1z7 - 2 + 6z2 - z4 - 15z6 + 12z8 - 4az + 18az3 - 24az5 + 7az9 - 2a2 + 24a2z4 - 40a2z6 + 13a2z8 + 2a2z10 - 3a3z + 8a3z3 + 7a3z5 - 25a3z7 + 11a3z9 + a4 - 9a4z2 + 29a4z4 - 27a4z6 + 4a4z8 + 2a4z10 - 3a5z + 5a5z3 + 5a5z5 - 11a5z7 + 4a5z9 + a6 - 6a6z2 + 13a6z4 - 11a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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=0 is the signature of 1167. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   6 1   j = 3                 8 3     j = 1               10 6       j = -1             11 9         j = -3           9 9           j = -5         7 11             j = -7       5 9               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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, 67]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 67]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[22, 13, 1, 14], X[20, 15, 21, 16], > X[10, 18, 11, 17], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 67]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 67]]
Out[5]=	DTCode[4, 8, 16, 12, 2, 18, 22, 20, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 67]][t]
Out[6]=	2 12 29 2 3 39 - -- + -- - -- - 29 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 67]][z]
Out[7]=	2 6 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 67], Knot[11, Alternating, 104], > Knot[11, Alternating, 168]}
In[9]:=	{KnotDet[Knot[11, Alternating, 67]], KnotSignature[Knot[11, Alternating, 67]]}
Out[9]=	{125, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 67]][q]
Out[10]=	-7 3 7 12 16 20 20 2 3 4 18 - q + -- - -- + -- - -- + -- - -- - 14 q + 9 q - 4 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 67], Knot[11, Alternating, 317]}
In[12]:=	A2Invariant[Knot[11, Alternating, 67]][q]
Out[12]=	-22 -18 3 2 -12 2 5 -6 2 2 4 -3 - q + q - --- + --- + q - --- + -- - q + -- + 3 q - 4 q + 16 14 10 8 4 q q q q q 6 8 10 12 > 2 q + 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 67]][a, z]
Out[13]=	2 4 -2 2 4 6 2 z 2 2 4 2 6 2 4 z -2 + a + 2 a + a - a - 3 z + -- + a z + 3 a z - a z - 2 z + -- - 2 2 a a 2 4 4 4 6 2 6 > a z + 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 67]][a, z]
Out[14]=	-2 2 4 6 2 z 3 5 7 2 -2 - a - 2 a + a + a - --- - 4 a z - 3 a z - 3 a z - 2 a z + 6 z + a 2 3 3 3 z 4 2 6 2 z 9 z 3 3 3 5 3 > ---- - 9 a z - 6 a z - -- + ---- + 18 a z + 8 a z + 5 a z + 2 3 a a a 4 4 5 5 7 3 4 z 8 z 2 4 4 4 6 4 4 z 18 z > 5 a z - z + -- - ---- + 24 a z + 29 a z + 13 a z + ---- - ----- - 4 2 3 a a a a 6 5 3 5 5 5 7 5 6 9 z 2 6 > 24 a z + 7 a z + 5 a z - 4 a z - 15 z + ---- - 40 a z - 2 a 7 4 6 6 6 13 z 3 7 5 7 7 7 8 > 27 a z - 11 a z + ----- - 25 a z - 11 a z + a z + 12 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 13 a z + 4 a z + 3 a z + 7 a z + 11 a z + 4 a z + 2 a z + 4 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 67]], Vassiliev[3][Knot[11, Alternating, 67]]}
Out[15]=	{1, -3}
In[16]:=	Kh[Knot[11, Alternating, 67]][q, t]
Out[16]=	9 1 2 1 5 2 7 5 9 - + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 7 11 9 9 11 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 8 q t + 3 q t + 6 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a67

K11a66 K11a68