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

K11n55 K11n57

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X9,16,10,17 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,21,18,20 X19,13,20,12 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -14 2 -16 -18 -6 -22 -20 -12 -10

Alexander Polynomial:

t-3 - 4t-2 + 8t-1 - 9 + 8t - 4t2 + t3

Conway Polynomial:

1 + z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{816, 10156, K11n15, K11n58, ...}

Determinant and Signature:

{35, 2}

Jones Polynomial:

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

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

{K11n58, ...}

A2 (sl(3)) Invariant:

- q-12 - q-8 + 2q-4 + q-2 + 3 + q2 - 2q8 - q12 + q16

HOMFLY-PT Polynomial:

a-4 + a-4z2 - 4a-2 - 6a-2z2 - 2a-2z4 + 6 + 9z2 + 5z4 + z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - 2a-5z + 3a-5z3 + a-4 - a-4z2 + a-4z6 - 7a-3z + 15a-3z3 - 11a-3z5 + 3a-3z7 + 4a-2 - 13a-2z2 + 17a-2z4 - 12a-2z6 + 3a-2z8 - 10a-1z + 24a-1z3 - 17a-1z5 + a-1z7 + a-1z9 + 6 - 20z2 + 32z4 - 23z6 + 5z8 - 8az + 19az3 - 11az5 - az7 + az9 + 2a2 - 9a2z2 + 15a2z4 - 10a2z6 + 2a2z8 - 3a3z + 7a3z3 - 5a3z5 + a3z7

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

{1, -1}

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 1156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   1 j = 9                 2   j = 7               2 1   j = 5             3 2     j = 3           3 2       j = 1         3 4         j = -1       2 2           j = -3     1 3             j = -5   1 2               j = -7   1                 j = -9 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, NonAlternating, 56]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 56]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[13, 6, 14, 7], X[15, 22, 16, 1], > X[17, 21, 18, 20], X[19, 13, 20, 12], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 56]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 56]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -18, -6, -22, -20, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 56]][t]
Out[6]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 56]][z]
Out[7]=	2 4 6 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], > Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 56]], KnotSignature[Knot[11, NonAlternating, 56]]}
Out[9]=	{35, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 56]][q]
Out[10]=	-4 2 3 5 2 3 4 5 -5 - q + -- - -- + - + 6 q - 5 q + 4 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 56]][q]
Out[12]=	-12 -8 2 -2 2 8 12 16 3 - q - q + -- + q + q - 2 q - q + q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 56]][a, z]
Out[13]=	2 2 4 -4 4 2 2 z 6 z 2 2 4 2 z 2 4 6 6 + a - -- - 2 a + 9 z + -- - ---- - 3 a z + 5 z - ---- - a z + z 2 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 56]][a, z]
Out[14]=	2 2 -4 4 2 2 z 7 z 10 z 3 2 z z 6 + a + -- + 2 a - --- - --- - ---- - 8 a z - 3 a z - 20 z + -- - -- - 2 5 3 a 6 4 a a a a a 2 3 3 3 13 z 2 2 3 z 15 z 24 z 3 3 3 4 > ----- - 9 a z + ---- + ----- + ----- + 19 a z + 7 a z + 32 z + 2 5 3 a a a a 4 5 5 6 6 17 z 2 4 11 z 17 z 5 3 5 6 z 12 z > ----- + 15 a z - ----- - ----- - 11 a z - 5 a z - 23 z + -- - ----- - 2 3 a 4 2 a a a a 7 7 8 9 2 6 3 z z 7 3 7 8 3 z 2 8 z 9 > 10 a z + ---- + -- - a z + a z + 5 z + ---- + 2 a z + -- + a z 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 56]], Vassiliev[3][Knot[11, NonAlternating, 56]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 56]][q, t]
Out[16]=	3 1 1 1 2 1 3 2 2 3 q 4 q + 3 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 11 4 > 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n56

K11n55 K11n57