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

K11a60 K11a62

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 10 2 18 20 6 22 14 12

Alexander Polynomial:

- 6t-2 + 26t-1 - 39 + 26t - 6t2

Conway Polynomial:

1 + 2z2 - 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

q-1 - 3 + 7q - 11q2 + 15q3 - 16q4 + 16q5 - 14q6 + 10q7 - 6q8 + 3q9 - q10

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

{...}

A2 (sl(3)) Invariant:

q-4 - q-2 - 1 + 4q2 - 2q4 + q6 + 3q8 - 2q10 + 2q12 - 2q14 - 3q20 + 4q22 - q26 + 2q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 2a-8 + 3a-8z2 - a-6 - a-6z2 - 2a-6z4 - a-4 - 3a-4z2 - 3a-4z4 + 2a-2 + 2a-2z2 - a-2z4 + z2

Kauffman Polynomial:

- 2a-11z + 5a-11z3 - 4a-11z5 + a-11z7 + a-10 - 6a-10z2 + 15a-10z4 - 12a-10z6 + 3a-10z8 - 3a-9z + 10a-9z3 - 2a-9z5 - 7a-9z7 + 3a-9z9 + 2a-8 - 15a-8z2 + 36a-8z4 - 32a-8z6 + 7a-8z8 + a-8z10 - 3a-7z + 8a-7z3 - 3a-7z5 - 13a-7z7 + 7a-7z9 + a-6 - 9a-6z2 + 24a-6z4 - 32a-6z6 + 11a-6z8 + a-6z10 - 3a-5z + 13a-5z3 - 19a-5z5 + 3a-5z7 + 4a-5z9 - a-4 + 7a-4z2 - 5a-4z4 - 6a-4z6 + 7a-4z8 - a-3z + 8a-3z3 - 11a-3z5 + 8a-3z7 - 2a-2 + 6a-2z2 - 7a-2z4 + 6a-2z6 - 2a-1z3 + 3a-1z5 - z2 + z4

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

{2, 5}

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 1161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 21                       1 j = 19                     2   j = 17                   4 1   j = 15                 6 2     j = 13               8 4       j = 11             8 6         j = 9           8 8           j = 7         7 8             j = 5       4 8               j = 3     3 7                 j = 1   1 5                   j = -1   2                     j = -3 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, 61]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 61]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 6, 17, 5], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 61]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 61]]
Out[5]=	DTCode[4, 8, 16, 10, 2, 18, 20, 6, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 61]][t]
Out[6]=	6 26 2 -39 - -- + -- + 26 t - 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 61]][z]
Out[7]=	2 4 1 + 2 z - 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 61]}
In[9]:=	{KnotDet[Knot[11, Alternating, 61]], KnotSignature[Knot[11, Alternating, 61]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 61]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -3 + - + 7 q - 11 q + 15 q - 16 q + 16 q - 14 q + 10 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 61]}
In[12]:=	A2Invariant[Knot[11, Alternating, 61]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 20 -1 + q - q + 4 q - 2 q + q + 3 q - 2 q + 2 q - 2 q - 3 q + 22 26 28 30 32 > 4 q - q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 61]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 2 -6 -4 2 2 3 z z 3 z 2 z 2 z 3 z z -a + -- - a - a + -- + z + ---- - -- - ---- + ---- - ---- - ---- - -- 8 2 8 6 4 2 6 4 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 61]][a, z]
Out[14]=	2 2 -10 2 -6 -4 2 2 z 3 z 3 z 3 z z 2 6 z 15 z a + -- + a - a - -- - --- - --- - --- - --- - -- - z - ---- - ----- - 8 2 11 9 7 5 3 10 8 a a a a a a a a a 2 2 2 3 3 3 3 3 3 9 z 7 z 6 z 5 z 10 z 8 z 13 z 8 z 2 z 4 > ---- + ---- + ---- + ---- + ----- + ---- + ----- + ---- - ---- + z + 6 4 2 11 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 15 z 36 z 24 z 5 z 7 z 4 z 2 z 3 z 19 z 11 z > ----- + ----- + ----- - ---- - ---- - ---- - ---- - ---- - ----- - ----- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 5 6 6 6 6 6 7 7 7 7 3 z 12 z 32 z 32 z 6 z 6 z z 7 z 13 z 3 z > ---- - ----- - ----- - ----- - ---- + ---- + --- - ---- - ----- + ---- + a 10 8 6 4 2 11 9 7 5 a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 8 z 3 z 7 z 11 z 7 z 3 z 7 z 4 z z z > ---- + ---- + ---- + ----- + ---- + ---- + ---- + ---- + --- + --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 61]], Vassiliev[3][Knot[11, Alternating, 61]]}
Out[15]=	{2, 5}
In[16]:=	Kh[Knot[11, Alternating, 61]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 8 q t + 7 q t + 8 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 8 q t + 8 q t + 8 q t + 6 q t + 8 q t + 4 q t + 6 q t + 15 7 17 7 17 8 19 8 21 9 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a61

K11a60 K11a62