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

K11a172 K11a174

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X20,6,21,5 X14,8,15,7 X2,10,3,9 X8,12,9,11 X18,13,19,14 X22,15,1,16 X6,18,7,17 X12,19,13,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 20 14 2 8 18 22 6 12 16

Alexander Polynomial:

2t-3 - 12t-2 + 32t-1 - 43 + 32t - 12t2 + 2t3

Conway Polynomial:

1 + 2z2 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 13 + 18q - 21q2 + 23q3 - 19q4 + 15q5 - 10q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 3q-2 - 2 + 3q2 - 4q4 + 4q6 + 2q10 + 4q12 - 3q14 + 4q16 - 3q18 - 3q20 + 2q22 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-9z3 + a-9z5 - 4a-8z4 + 4a-8z6 - 4a-7z + 9a-7z3 - 15a-7z5 + 9a-7z7 + 2a-6 - 6a-6z2 + 15a-6z4 - 20a-6z6 + 11a-6z8 - 6a-5z + 13a-5z3 - 11a-5z5 - 3a-5z7 + 7a-5z9 + 3a-4 - 5a-4z2 + 21a-4z4 - 31a-4z6 + 13a-4z8 + 2a-4z10 - 4a-3z + 4a-3z3 + 5a-3z5 - 17a-3z7 + 11a-3z9 - a-2 + 6a-2z2 - 13a-2z6 + 6a-2z8 + 2a-2z10 - 4a-1z + 8a-1z3 - 8a-1z5 - 2a-1z7 + 4a-1z9 - 2 + 8z2 - 5z4 - 5z6 + 4z8 - 2az + 7az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{2, 5}

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 11173. 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 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   7 1   j = 11                 8 3     j = 9               11 7       j = 7             12 8         j = 5           9 11           j = 3         9 12             j = 1       5 10               j = -1     2 8                 j = -3   1 5                   j = -5   2                     j = -7 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, 173]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 173]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[20, 6, 21, 5], X[14, 8, 15, 7], > X[2, 10, 3, 9], X[8, 12, 9, 11], X[18, 13, 19, 14], X[22, 15, 1, 16], > X[6, 18, 7, 17], X[12, 19, 13, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 173]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -10, 7, -4, 8, -11, 9, -7, 10, > -3, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 173]]
Out[5]=	DTCode[4, 10, 20, 14, 2, 8, 18, 22, 6, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 173]][t]
Out[6]=	2 12 32 2 3 -43 + -- - -- + -- + 32 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 173]][z]
Out[7]=	2 6 1 + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 173]}
In[9]:=	{KnotDet[Knot[11, Alternating, 173]], KnotSignature[Knot[11, Alternating, 173]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 173]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -13 + q - -- + - + 18 q - 21 q + 23 q - 19 q + 15 q - 10 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 173]}
In[12]:=	A2Invariant[Knot[11, Alternating, 173]][q]
Out[12]=	-10 -6 3 3 2 4 6 10 12 14 -2 + q - q + -- - -- + 3 q - 4 q + 4 q + 2 q + 4 q - 3 q + 4 2 q q 16 18 20 22 24 > 4 q - 3 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 173]][a, z]
Out[13]=	2 2 2 4 4 2 3 -2 2 2 z 4 z z 2 2 4 z 2 z -2 - -- + -- + a + a - 3 z - -- + ---- + -- + a z - 2 z - -- + ---- + 6 4 6 4 2 6 4 a a a a a a a 4 6 6 z z z > -- + -- + -- 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 173]][a, z]
Out[14]=	2 2 2 3 -2 2 4 z 6 z 4 z 4 z 2 6 z 5 z -2 + -- + -- - a - a - --- - --- - --- - --- - 2 a z + 8 z - ---- - ---- + 6 4 7 5 3 a 6 4 a a a a a a a 2 3 3 3 3 3 4 6 z 2 2 z 9 z 13 z 4 z 8 z 3 4 4 z > ---- + 3 a z - -- + ---- + ----- + ---- + ---- + 7 a z - 5 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 5 5 5 5 5 15 z 21 z 2 4 z 15 z 11 z 5 z 8 z 5 > ----- + ----- - 3 a z + -- - ----- - ----- + ---- - ---- - 8 a z - 6 4 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 6 4 z 20 z 31 z 13 z 2 6 9 z 3 z 17 z 2 z > 5 z + ---- - ----- - ----- - ----- + a z + ---- - ---- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 11 z 13 z 6 z 7 z 11 z 4 z 2 z 2 z > 3 a z + 4 z + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 173]], Vassiliev[3][Knot[11, Alternating, 173]]}
Out[15]=	{2, 5}
In[16]:=	Kh[Knot[11, Alternating, 173]][q, t]
Out[16]=	3 1 2 1 5 2 8 5 q 3 10 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 12 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 9 q t + 11 q t + 12 q t + 8 q t + 11 q t + 7 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a173

K11a172 K11a174