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

K11a55 K11a57

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 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, -5, 10, -6, 11, -7}

DT (Dowker-Thistlethwaite) Code:

4 8 16 2 18 20 22 6 14 12 10

Alexander Polynomial:

- 2t-3 + 11t-2 - 25t-1 + 33 - 25t + 11t2 - 2t3

Conway Polynomial:

1 + z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a185, K11a265, ...}

Determinant and Signature:

{109, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 11q-3 + 15q-2 - 17q-1 + 18 - 15q + 12q2 - 7q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + q-14 + q-12 - 4q-10 + 2q-8 - 2q-6 - q-4 + 3q-2 - 1 + 5q2 - q4 + q6 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + 3a-3z3 - 7a-3z5 + 5a-3z7 - a-2 + 4a-2z2 - 4a-2z4 - 3a-2z6 + 5a-2z8 - 3a-1z + 11a-1z3 - 15a-1z5 + 5a-1z7 + 3a-1z9 + 3 - 4z2 + 6z4 - 13z6 + 8z8 + z10 - 5az + 10az3 - 8az5 - 4az7 + 6az9 + 3a2 - 8a2z2 + 13a2z4 - 17a2z6 + 7a2z8 + a2z10 - 5a3z + 12a3z3 - 10a3z5 - a3z7 + 3a3z9 + a4 - a4z2 + 5a4z4 - 9a4z6 + 4a4z8 - 2a5z + 8a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{1, 2}

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 1156. 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                     2   j = 7                   5 1   j = 5                 7 2     j = 3               8 5       j = 1             10 7         j = -1           8 9           j = -3         7 9             j = -5       4 8               j = -7     2 7                 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, 56]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 56]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[16, 6, 17, 5], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[22, 13, 1, 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, 56]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -5, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 56]]
Out[5]=	DTCode[4, 8, 16, 2, 18, 20, 22, 6, 14, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 56]][t]
Out[6]=	2 11 25 2 3 33 - -- + -- - -- - 25 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 56]][z]
Out[7]=	2 4 6 1 + z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 56], Knot[11, Alternating, 185], > Knot[11, Alternating, 265]}
In[9]:=	{KnotDet[Knot[11, Alternating, 56]], KnotSignature[Knot[11, Alternating, 56]]}
Out[9]=	{109, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 56]][q]
Out[10]=	-6 3 6 11 15 17 2 3 4 5 18 + q - -- + -- - -- + -- - -- - 15 q + 12 q - 7 q + 3 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, 56]}
In[12]:=	A2Invariant[Knot[11, Alternating, 56]][q]
Out[12]=	-18 -16 -14 -12 4 2 2 -4 3 2 4 6 -1 + q - q + q + q - --- + -- - -- - q + -- + 5 q - q + q + 10 8 6 2 q q q q 8 10 12 16 > 2 q - 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 56]][a, z]
Out[13]=	2 2 4 -4 -2 2 4 2 z 3 z 2 2 4 2 4 2 z 3 - a + a - 3 a + a + 2 z - -- + ---- - 5 a z + 2 a z - z + ---- - 4 2 2 a a a 2 4 4 4 6 2 6 > 3 a z + a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 56]][a, z]
Out[14]=	2 -4 -2 2 4 z 3 z 3 5 2 3 z 3 - a - a + 3 a + a + -- - --- - 5 a z - 5 a z - 2 a z - 4 z + ---- + 5 a 4 a a 2 3 3 3 4 z 2 2 4 2 6 2 2 z 3 z 11 z 3 > ---- - 8 a z - a z + 2 a z - ---- + ---- + ----- + 10 a z + 2 5 3 a a a a 4 4 3 3 5 3 4 5 z 4 z 2 4 4 4 6 4 > 12 a z + 8 a z + 6 z - ---- - ---- + 13 a z + 5 a z - 3 a z + 4 2 a a 5 5 5 6 6 z 7 z 15 z 5 3 5 5 5 6 3 z 3 z > -- - ---- - ----- - 8 a z - 10 a z - 9 a z - 13 z + ---- - ---- - 5 3 a 4 2 a a a a 7 7 2 6 4 6 6 6 5 z 5 z 7 3 7 5 7 > 17 a z - 9 a z + a z + ---- + ---- - 4 a z - a z + 3 a z + 3 a a 8 9 8 5 z 2 8 4 8 3 z 9 3 9 10 2 10 > 8 z + ---- + 7 a z + 4 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 56]], Vassiliev[3][Knot[11, Alternating, 56]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 56]][q, t]
Out[16]=	9 1 2 1 4 2 7 4 8 - + 10 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 7 9 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 8 q t + 5 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a56

K11a55 K11a57