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

K11a58 K11a60

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X16,6,17,5 X2837 X22,9,1,10 X20,11,21,12 X18,13,19,14 X6,16,7,15 X14,17,15,18 X12,19,13,20 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 2 22 20 18 6 14 12 10

Alexander Polynomial:

- 3t-2 + 11t-1 - 15 + 11t - 3t2

Conway Polynomial:

1 - z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{107, K11n3, ...}

Determinant and Signature:

{43, 2}

Jones Polynomial:

q-5 - 2q-4 + 3q-3 - 4q-2 + 5q-1 - 6 + 6q - 5q2 + 5q3 - 3q4 + 2q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-10 - q-8 - q-2 + q6 + 2q10 + q12 + q16 - q18 - q20

HOMFLY-PT Polynomial:

- a-6 + 2a-4 + 2a-4z2 - a-2z2 - a-2z4 - z2 - z4 - a2 - 2a2z2 - a2z4 + a4 + a4z2

Kauffman Polynomial:

- a-7z + a-7z3 + a-6 - 2a-6z2 + 2a-6z4 - a-5z + 2a-5z5 + 2a-4 - 3a-4z2 + 2a-4z6 - a-3z3 - 2a-3z5 + 2a-3z7 - 4a-2z6 + 2a-2z8 + a-1z - 5a-1z3 + 7a-1z5 - 7a-1z7 + 2a-1z9 - 3z2 + 9z4 - 5z6 - 2z8 + z10 + 4az - 19az3 + 33az5 - 21az7 + 4az9 + a2 - 11a2z2 + 18a2z4 - 5a2z6 - 3a2z8 + a2z10 + 3a3z - 14a3z3 + 22a3z5 - 12a3z7 + 2a3z9 + a4 - 7a4z2 + 11a4z4 - 6a4z6 + a4z8

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=2 is the signature of 1159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                       1 j = 11                     1   j = 9                   2 1   j = 7                 3 1     j = 5               2 2       j = 3             4 3         j = 1           3 3           j = -1         2 3             j = -3       2 3               j = -5     1 2                 j = -7   1 2                   j = -9   1                     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, 59]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 59]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[22, 9, 1, 10], X[20, 11, 21, 12], X[18, 13, 19, 14], X[6, 16, 7, 15], > X[14, 17, 15, 18], X[12, 19, 13, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 59]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -7, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 59]]
Out[5]=	DTCode[4, 8, 16, 2, 22, 20, 18, 6, 14, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 59]][t]
Out[6]=	3 11 2 -15 - -- + -- + 11 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 59]][z]
Out[7]=	2 4 1 - z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 7], Knot[11, Alternating, 59], Knot[11, NonAlternating, 3]}
In[9]:=	{KnotDet[Knot[11, Alternating, 59]], KnotSignature[Knot[11, Alternating, 59]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 59]][q]
Out[10]=	-5 2 3 4 5 2 3 4 5 6 -6 + q - -- + -- - -- + - + 6 q - 5 q + 5 q - 3 q + 2 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, 59]}
In[12]:=	A2Invariant[Knot[11, Alternating, 59]][q]
Out[12]=	-16 -10 -8 -2 6 10 12 16 18 20 q + q - q - q + q + 2 q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 59]][a, z]
Out[13]=	2 2 4 -6 2 2 4 2 2 z z 2 2 4 2 4 z 2 4 -a + -- - a + a - z + ---- - -- - 2 a z + a z - z - -- - a z 4 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 59]][a, z]
Out[14]=	2 2 -6 2 2 4 z z z 3 2 2 z 3 z a + -- + a + a - -- - -- + - + 4 a z + 3 a z - 3 z - ---- - ---- - 4 7 5 a 6 4 a a a a a 3 3 3 4 2 2 4 2 z z 5 z 3 3 3 4 2 z > 11 a z - 7 a z + -- - -- - ---- - 19 a z - 14 a z + 9 z + ---- + 7 3 a 6 a a a 5 5 5 2 4 4 4 2 z 2 z 7 z 5 3 5 6 > 18 a z + 11 a z + ---- - ---- + ---- + 33 a z + 22 a z - 5 z + 5 3 a a a 6 6 7 7 2 z 4 z 2 6 4 6 2 z 7 z 7 3 7 8 > ---- - ---- - 5 a z - 6 a z + ---- - ---- - 21 a z - 12 a z - 2 z + 4 2 3 a a a a 8 9 2 z 2 8 4 8 2 z 9 3 9 10 2 10 > ---- - 3 a z + a z + ---- + 4 a z + 2 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 59]], Vassiliev[3][Knot[11, Alternating, 59]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 59]][q, t]
Out[16]=	3 1 1 1 2 1 2 2 3 3 q + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 2 3 3 q 3 5 5 2 7 2 7 3 9 3 > ---- + --- + --- + 3 q t + 2 q t + 2 q t + 3 q t + q t + 2 q t + 2 q t t q t 9 4 11 4 13 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a59

K11a58 K11a60