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

K11a45 K11a47

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 20 18 16 6 12 22 10

Alexander Polynomial:

2t-3 - 9t-2 + 20t-1 - 25 + 20t - 9t2 + 2t3

Conway Polynomial:

1 + 2z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1084, K11n184, ...}

Determinant and Signature:

{87, 2}

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 9q-1 - 11 + 14q - 13q2 + 12q3 - 9q4 + 5q5 - 3q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 - q-6 + 2q-4 - 2q-2 + 2 + 2q2 + q4 + 4q6 - q8 + q10 - 2q12 - 3q14 + q16 - q18 + q22

HOMFLY-PT Polynomial:

a-6 + a-6z2 - 4a-4 - 5a-4z2 - 2a-4z4 + 4a-2 + 5a-2z2 + 3a-2z4 + a-2z6 + 1 + 3z2 + 3z4 + z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 + 2a-7z - 4a-7z3 + 3a-7z5 - a-6 + a-6z2 - 3a-6z4 + 4a-6z6 + 6a-5z - 9a-5z3 + a-5z5 + 4a-5z7 - 4a-4 + 8a-4z2 - 7a-4z4 - a-4z6 + 4a-4z8 + 4a-3z - 2a-3z3 - 4a-3z5 - a-3z7 + 3a-3z9 - 4a-2 + 3a-2z2 + 12a-2z4 - 20a-2z6 + 6a-2z8 + a-2z10 - 2a-1z + 8a-1z3 + a-1z5 - 14a-1z7 + 6a-1z9 + 1 - 9z2 + 29z4 - 27z6 + 5z8 + z10 - 4az + 10az3 - az5 - 8az7 + 3az9 + a2 - 6a2z2 + 14a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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

{2, 0}

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 1146. 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 r = 5 r = 6 j = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 6 2     j = 7               6 3       j = 5             7 6         j = 3           7 6           j = 1         5 8             j = -1       4 6               j = -3     2 5                 j = -5   1 4                   j = -7   2                     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, Alternating, 46]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 46]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[18, 11, 19, 12], X[16, 13, 17, 14], X[6, 15, 7, 16], > X[12, 17, 13, 18], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 46]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -7, 9, -6, 10, > -5, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 46]]
Out[5]=	DTCode[4, 8, 14, 2, 20, 18, 16, 6, 12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 46]][t]
Out[6]=	2 9 20 2 3 -25 + -- - -- + -- + 20 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 46]][z]
Out[7]=	2 4 6 1 + 2 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 84], Knot[11, Alternating, 46], Knot[11, NonAlternating, 184]}
In[9]:=	{KnotDet[Knot[11, Alternating, 46]], KnotSignature[Knot[11, Alternating, 46]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 46]][q]
Out[10]=	-4 3 6 9 2 3 4 5 6 7 -11 - q + -- - -- + - + 14 q - 13 q + 12 q - 9 q + 5 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, Alternating, 46]}
In[12]:=	A2Invariant[Knot[11, Alternating, 46]][q]
Out[12]=	-12 -10 -8 -6 2 2 2 4 6 8 10 12 2 - q + q - q - q + -- - -- + 2 q + q + 4 q - q + q - 2 q - 4 2 q q 14 16 18 22 > 3 q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 46]][a, z]
Out[13]=	2 2 2 4 -6 4 4 2 2 z 5 z 5 z 2 2 4 2 z 1 + a - -- + -- - a + 3 z + -- - ---- + ---- - 2 a z + 3 z - ---- + 4 2 6 4 2 4 a a a a a a 4 6 3 z 2 4 6 z > ---- - a z + z + -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 46]][a, z]
Out[14]=	2 -6 4 4 2 2 z 6 z 4 z 2 z 3 2 z 1 - a - -- - -- + a + --- + --- + --- - --- - 4 a z - 2 a z - 9 z - -- + 4 2 7 5 3 a 8 a a a a a a 2 2 2 3 3 3 3 z 8 z 3 z 2 2 4 z 9 z 2 z 8 z 3 > -- + ---- + ---- - 6 a z - ---- - ---- - ---- + ---- + 10 a z + 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 5 5 3 3 4 z 3 z 7 z 12 z 2 4 3 z z 4 z > 5 a z + 29 z + -- - ---- - ---- + ----- + 14 a z + ---- + -- - ---- + 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 7 7 z 5 3 5 6 4 z z 20 z 2 6 4 z z > -- - a z - 4 a z - 27 z + ---- - -- - ----- - 12 a z + ---- - -- - a 6 4 2 5 3 a a a a a 7 8 8 9 9 14 z 7 3 7 8 4 z 6 z 2 8 3 z 6 z > ----- - 8 a z + a z + 5 z + ---- + ---- + 3 a z + ---- + ---- + a 4 2 3 a a a a 10 9 10 z > 3 a z + z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 46]], Vassiliev[3][Knot[11, Alternating, 46]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, Alternating, 46]][q, t]
Out[16]=	3 1 2 1 4 2 5 4 6 5 q 8 q + 7 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 9 4 > 6 q t + 7 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 11 4 11 5 13 5 15 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a46

K11a45 K11a47