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

K11a64 K11a66

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 10 2 22 20 18 6 14 12

Alexander Polynomial:

- 3t-2 + 15t-1 - 23 + 15t - 3t2

Conway Polynomial:

1 + 3z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{59, -2}

Jones Polynomial:

- q-10 + 2q-9 - 4q-8 + 6q-7 - 7q-6 + 9q-5 - 9q-4 + 8q-3 - 6q-2 + 4q-1 - 2 + q

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 - q-26 + 2q-22 - q-20 + q-18 + q-16 + q-12 - q-10 + q-8 - 2q-4 + 2q-2 + q4

HOMFLY-PT Polynomial:

1 + z2 - a2 - a2z2 - a2z4 + a4 + a4z2 - a4z4 - a6z4 + a8 + 2a8z2 - a10

Kauffman Polynomial:

1 - 2z2 + z4 + az - 3az3 + 2az5 + a2 - 3a2z2 + 2a2z6 + a3z - 2a3z3 + 2a3z7 + a4 - 2a4z2 + 2a4z4 - 2a4z6 + 2a4z8 - 4a5z3 + 7a5z5 - 5a5z7 + 2a5z9 + a6z2 + a6z4 - 2a6z6 - a6z8 + a6z10 + a7z - 8a7z3 + 17a7z5 - 15a7z7 + 4a7z9 + a8 - 3a8z2 + 9a8z4 - 7a8z6 - a8z8 + a8z10 - 2a9z + 4a9z3 + 3a9z5 - 7a9z7 + 2a9z9 + a10 - 5a10z2 + 11a10z4 - 9a10z6 + 2a10z8 - 3a11z + 7a11z3 - 5a11z5 + a11z7

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

{3, -9}

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 1165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     1   j = -1                   3 1   j = -3                 4 2     j = -5               4 2       j = -7             5 4         j = -9           4 4           j = -11         3 5             j = -13       3 4               j = -15     1 3                 j = -17   1 3                   j = -19   1                     j = -21 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, 65]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 65]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[22, 11, 1, 12], X[20, 13, 21, 14], X[18, 15, 19, 16], > X[6, 17, 7, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 65]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 65]]
Out[5]=	DTCode[4, 8, 16, 10, 2, 22, 20, 18, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 65]][t]
Out[6]=	3 15 2 -23 - -- + -- + 15 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 65]][z]
Out[7]=	2 4 1 + 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 65]}
In[9]:=	{KnotDet[Knot[11, Alternating, 65]], KnotSignature[Knot[11, Alternating, 65]]}
Out[9]=	{59, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 65]][q]
Out[10]=	-10 2 4 6 7 9 9 8 6 4 -2 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q 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, 65]}
In[12]:=	A2Invariant[Knot[11, Alternating, 65]][q]
Out[12]=	-32 -30 -28 -26 2 -20 -18 -16 -12 -10 -8 -q - q + q - q + --- - q + q + q + q - q + q - 22 q 2 2 4 > -- + -- + q 4 2 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 65]][a, z]
Out[13]=	2 4 8 10 2 2 2 4 2 8 2 2 4 4 4 6 4 1 - a + a + a - a + z - a z + a z + 2 a z - a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 65]][a, z]
Out[14]=	2 4 8 10 3 7 9 11 2 1 + a + a + a + a + a z + a z + a z - 2 a z - 3 a z - 2 z - 2 2 4 2 6 2 8 2 10 2 3 3 3 > 3 a z - 2 a z + a z - 3 a z - 5 a z - 3 a z - 2 a z - 5 3 7 3 9 3 11 3 4 4 4 6 4 8 4 > 4 a z - 8 a z + 4 a z + 7 a z + z + 2 a z + a z + 9 a z + 10 4 5 5 5 7 5 9 5 11 5 2 6 > 11 a z + 2 a z + 7 a z + 17 a z + 3 a z - 5 a z + 2 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 2 a z - 2 a z - 7 a z - 9 a z + 2 a z - 5 a z - 15 a z - 9 7 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 7 a z + a z + 2 a z - a z - a z + 2 a z + 2 a z + 4 a z + 9 9 6 10 8 10 > 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 65]], Vassiliev[3][Knot[11, Alternating, 65]]}
Out[15]=	{3, -9}
In[16]:=	Kh[Knot[11, Alternating, 65]][q, t]
Out[16]=	2 3 1 1 1 3 1 3 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 4 3 5 4 4 5 4 4 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 4 t 3 2 > ---- + - + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a65

K11a64 K11a66