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

K11a65 K11a67

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 X20,13,21,14 X22,15,1,16 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, -7, 11, -8}

DT (Dowker-Thistlethwaite) Code:

4 8 16 12 2 18 20 22 10 6 14

Alexander Polynomial:

- t-4 + 6t-3 - 15t-2 + 24t-1 - 27 + 24t - 15t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 + z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a163, ...}

Determinant and Signature:

{119, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 16q-4 + 19q-3 - 19q-2 + 17q-1 - 12 + 8q - 4q2 + q3

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

{K11a121, ...}

A2 (sl(3)) Invariant:

- q-24 - 2q-18 + 3q-16 - 2q-14 + 2q-12 + 2q-10 - 2q-8 + 4q-6 - 4q-4 + 3q-2 - q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial:

1 + 2z2 + 3z4 + z6 - 2a2 - 7a2z2 - 9a2z4 - 5a2z6 - a2z8 + 4a4 + 10a4z2 + 8a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 4a-1z3 - 10a-1z5 + 4a-1z7 + 1 - 5z2 + 16z4 - 20z6 + 7z8 + az - az3 + 6az5 - 13az7 + 6az9 + 2a2 - 17a2z2 + 41a2z4 - 35a2z6 + 8a2z8 + 2a2z10 + 2a3z - 12a3z3 + 29a3z5 - 28a3z7 + 11a3z9 + 4a4 - 21a4z2 + 36a4z4 - 26a4z6 + 7a4z8 + 2a4z10 - 3a5z3 + 5a5z5 - 6a5z7 + 5a5z9 + 2a6 - 7a6z2 + 8a6z4 - 9a6z6 + 6a6z8 + 2a7z3 - 7a7z5 + 5a7z7 + 2a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{2, -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 1166. 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 = 7                       1 j = 5                     3   j = 3                   5 1   j = 1                 7 3     j = -1               10 5       j = -3             10 8         j = -5           9 9           j = -7         7 10             j = -9       5 9               j = -11     2 7                 j = -13   1 5                   j = -15   2                     j = -17 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, 66]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 66]]
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[20, 13, 21, 14], X[22, 15, 1, 16], > X[10, 18, 11, 17], X[6, 19, 7, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 66]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 66]]
Out[5]=	DTCode[4, 8, 16, 12, 2, 18, 20, 22, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 66]][t]
Out[6]=	-4 6 15 24 2 3 4 -27 - t + -- - -- + -- + 24 t - 15 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 66]][z]
Out[7]=	2 4 6 8 1 + 2 z + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 163]}
In[9]:=	{KnotDet[Knot[11, Alternating, 66]], KnotSignature[Knot[11, Alternating, 66]]}
Out[9]=	{119, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 66]][q]
Out[10]=	-8 3 7 12 16 19 19 17 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 121]}
In[12]:=	A2Invariant[Knot[11, Alternating, 66]][q]
Out[12]=	-24 2 3 2 2 2 2 4 4 3 2 4 6 8 -q - --- + --- - --- + --- + --- - -- + -- - -- + -- - q + 2 q - 2 q + q 18 16 14 12 10 8 6 4 2 q q q q q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 66]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 1 - 2 a + 4 a - 2 a + 2 z - 7 a z + 10 a z - 3 a z + 3 z - 9 a z + 4 4 6 4 6 2 6 4 6 2 8 > 8 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 66]][a, z]
Out[14]=	2 4 6 3 9 2 2 2 4 2 1 + 2 a + 4 a + 2 a + a z + 2 a z + a z - 5 z - 17 a z - 21 a z - 3 6 2 8 2 4 z 3 3 3 5 3 7 3 9 3 > 7 a z + 2 a z + ---- - a z - 12 a z - 3 a z + 2 a z - 2 a z + a 4 5 4 2 z 2 4 4 4 6 4 8 4 10 z 5 > 16 z - ---- + 41 a z + 36 a z + 8 a z - 5 a z - ----- + 6 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 29 a z + 5 a z - 7 a z + a z - 20 z + -- - 35 a z - 26 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 9 a z + 3 a z + ---- - 13 a z - 28 a z - 6 a z + 5 a z + 7 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 8 a z + 7 a z + 6 a z + 6 a z + 11 a z + 5 a z + 2 a z + 4 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 66]], Vassiliev[3][Knot[11, Alternating, 66]]}
Out[15]=	{2, -4}
In[16]:=	Kh[Knot[11, Alternating, 66]][q, t]
Out[16]=	8 10 1 2 1 5 2 7 5 9 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 10 9 9 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a66

K11a65 K11a67