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

K11a57 K11a59

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X16,6,17,5 X2837 X20,9,21,10 X22,11,1,12 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, -5, 11, -6}

DT (Dowker-Thistlethwaite) Code:

4 8 16 2 20 22 18 6 14 12 10

Alexander Polynomial:

- 2t-3 + 9t-2 - 18t-1 + 23 - 18t + 9t2 - 2t3

Conway Polynomial:

1 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1087, 1098, K11a165, K11n72, ...}

Determinant and Signature:

{81, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 9q-3 + 11q-2 - 13q-1 + 13 - 10q + 8q2 - 4q3 + 2q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 + q-12 - 2q-10 + 2q-8 - 2q-6 - q-4 - 2 + 3q2 + 3q6 + 3q8 - q10 - q14 - q16

HOMFLY-PT Polynomial:

- 2a-4 - a-4z2 + 5a-2 + 6a-2z2 + 2a-2z4 - 2 - 4z2 - 3z4 - z6 - a2 - 3a2z2 - 3a2z4 - a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

2a-5z - 3a-5z3 + a-5z5 - 2a-4 + 4a-4z2 - 5a-4z4 + 2a-4z6 + 4a-3z - 5a-3z3 - a-3z5 + 2a-3z7 - 5a-2 + 9a-2z2 - 6a-2z4 + 2a-2z8 + 5a-1z - 8a-1z3 + 7a-1z5 - 3a-1z7 + 2a-1z9 - 2 + 2z2 + 4z4 - 3z6 + z8 + z10 + 5az - 14az3 + 20az5 - 14az7 + 5az9 + a2 - 8a2z2 + 15a2z4 - 13a2z6 + 3a2z8 + a2z10 + a3z - 2a3z3 + 2a3z5 - 6a3z7 + 3a3z9 + a4 - 3a4z2 + 7a4z4 - 11a4z6 + 4a4z8 - a5z + 6a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{0, 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 1158. 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 = 11                       1 j = 9                     1   j = 7                   3 1   j = 5                 5 1     j = 3               5 3       j = 1             8 5         j = -1           6 6           j = -3         5 7             j = -5       4 6               j = -7     2 5                 j = -9   1 4                   j = -11   2                     j = -13 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, 58]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 58]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[20, 9, 21, 10], X[22, 11, 1, 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, 58]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -7, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 58]]
Out[5]=	DTCode[4, 8, 16, 2, 20, 22, 18, 6, 14, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 58]][t]
Out[6]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 58]][z]
Out[7]=	4 6 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[9]:=	{KnotDet[Knot[11, Alternating, 58]], KnotSignature[Knot[11, Alternating, 58]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 58]][q]
Out[10]=	-6 3 6 9 11 13 2 3 4 5 13 + q - -- + -- - -- + -- - -- - 10 q + 8 q - 4 q + 2 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 58]}
In[12]:=	A2Invariant[Knot[11, Alternating, 58]][q]
Out[12]=	-18 -16 -14 -12 2 2 2 -4 2 6 8 -2 + q - q + q + q - --- + -- - -- - q + 3 q + 3 q + 3 q - 10 8 6 q q q 10 14 16 > q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 58]][a, z]
Out[13]=	2 2 4 2 5 2 4 2 z 6 z 2 2 4 2 4 2 z -2 - -- + -- - a + a - 4 z - -- + ---- - 3 a z + 2 a z - 3 z + ---- - 4 2 4 2 2 a a a a a 2 4 4 4 6 2 6 > 3 a z + a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 58]][a, z]
Out[14]=	2 2 5 2 4 2 z 4 z 5 z 3 5 2 4 z -2 - -- - -- + a + a + --- + --- + --- + 5 a z + a z - a z + 2 z + ---- + 4 2 5 3 a 4 a a a a a 2 3 3 3 9 z 2 2 4 2 6 2 3 z 5 z 8 z 3 > ---- - 8 a z - 3 a z + 2 a z - ---- - ---- - ---- - 14 a z - 2 5 3 a a a a 4 4 3 3 5 3 4 5 z 6 z 2 4 4 4 6 4 > 2 a z + 6 a z + 4 z - ---- - ---- + 15 a z + 7 a z - 3 a z + 4 2 a a 5 5 5 6 z z 7 z 5 3 5 5 5 6 2 z 2 6 > -- - -- + ---- + 20 a z + 2 a z - 9 a z - 3 z + ---- - 13 a z - 5 3 a 4 a a a 7 7 8 4 6 6 6 2 z 3 z 7 3 7 5 7 8 2 z > 11 a z + a z + ---- - ---- - 14 a z - 6 a z + 3 a z + z + ---- + 3 a 2 a a 9 2 8 4 8 2 z 9 3 9 10 2 10 > 3 a z + 4 a z + ---- + 5 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 58]], Vassiliev[3][Knot[11, Alternating, 58]]}
Out[15]=	{0, 3}
In[16]:=	Kh[Knot[11, Alternating, 58]][q, t]
Out[16]=	6 1 2 1 4 2 5 4 6 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 7 6 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 5 q t + 3 q t + 5 q t + q t + 3 q t + 3 2 3 q t q t q t 7 4 9 4 11 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a58

K11a57 K11a59