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

K11n61 K11n63

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,20,12,21 X13,18,14,19 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-2 - 8t-1 + 13 - 8t + 2t2

Conway Polynomial:

1 + 2z4

Other knots with the same Alexander/Conway Polynomial:

{10146, K11n18, ...}

Determinant and Signature:

{33, 0}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 - q-6 - 2q-4 + q-2 + 2q2 + q4 + q6 + 2q8 - q10 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 2a-2 + 2a-2z2 + a-2z4 + 1 + z2 + z4 - 2a2 - 2a2z2 + a4

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 - a-4 + 3a-4z2 - 6a-4z4 + 2a-4z6 + a-3z + a-3z3 - 5a-3z5 + 2a-3z7 - 2a-2 - a-2z2 + 8a-2z4 - 7a-2z6 + 2a-2z8 + a-1z + a-1z3 + 3a-1z5 - 3a-1z7 + a-1z9 + 1 - 12z2 + 25z4 - 14z6 + 3z8 + 3az - 8az3 + 11az5 - 5az7 + az9 + 2a2 - 11a2z2 + 12a2z4 - 5a2z6 + a2z8 + 2a3z - 5a3z3 + 2a3z5 + a4 - 3a4z2 + a4z4

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

{0, 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 1162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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               2 1   j = 5             3 1     j = 3           2 2       j = 1         4 3         j = -1       2 3           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, 62]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 62]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 20, 12, 21], X[13, 18, 14, 19], X[6, 15, 7, 16], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 62]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 7, -10, > 6, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 62]]
Out[5]=	DTCode[4, 8, 14, 2, -16, -20, -18, 6, -22, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 62]][t]
Out[6]=	2 8 2 13 + -- - - - 8 t + 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 62]][z]
Out[7]=	4 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 146], Knot[11, NonAlternating, 18], Knot[11, NonAlternating, 62]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 62]], KnotSignature[Knot[11, NonAlternating, 62]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 62]][q]
Out[10]=	-4 2 3 5 2 3 4 5 6 + q - -- + -- - - - 5 q + 5 q - 3 q + 2 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, 62]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 62]][q]
Out[12]=	-14 -12 -10 -6 2 -2 2 4 6 8 10 16 q + q - q - q - -- + q + 2 q + q + q + 2 q - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 62]][a, z]
Out[13]=	2 2 4 -4 2 2 4 2 z 2 z 2 2 4 z 1 - a + -- - 2 a + a + z - -- + ---- - 2 a z + z + -- 2 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 62]][a, z]
Out[14]=	2 2 -4 2 2 4 z z z 3 2 3 z z 1 - a - -- + 2 a + a + -- + -- + - + 3 a z + 2 a z - 12 z + ---- - -- - 2 5 3 a 4 2 a a a a a 3 3 3 4 2 2 4 2 3 z z z 3 3 3 4 6 z > 11 a z - 3 a z - ---- + -- + -- - 8 a z - 5 a z + 25 z - ---- + 5 3 a 4 a a a 4 5 5 5 8 z 2 4 4 4 z 5 z 3 z 5 3 5 6 > ---- + 12 a z + a z + -- - ---- + ---- + 11 a z + 2 a z - 14 z + 2 5 3 a a a a 6 6 7 7 8 9 2 z 7 z 2 6 2 z 3 z 7 8 2 z 2 8 z > ---- - ---- - 5 a z + ---- - ---- - 5 a z + 3 z + ---- + a z + -- + 4 2 3 a 2 a a a a a 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 62]], Vassiliev[3][Knot[11, NonAlternating, 62]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 62]][q, t]
Out[16]=	3 1 1 1 2 1 3 2 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n62

K11n61 K11n63