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

K11a173 K11a175

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 2 22 6 20 8 10

Alexander Polynomial:

- t-4 + 5t-3 - 11t-2 + 15t-1 - 15 + 15t - 11t2 + 5t3 - t4

Conway Polynomial:

1 - z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{79, -2}

Jones Polynomial:

q-7 - 3q-6 + 5q-5 - 8q-4 + 11q-3 - 12q-2 + 12q-1 - 10 + 8q - 5q2 + 3q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 - q-14 - q-12 + q-10 - 2q-8 + 3q-6 - q-4 + q-2 + 1 - q2 + 2q4 + q8 - q12

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - a-2z4 + 4 + 11z2 + 9z4 + 2z6 - 3a2 - 12a2z2 - 13a2z4 - 6a2z6 - a2z8 + a4 + 4a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- a-3z + 4a-3z3 - 4a-3z5 + a-3z7 + a-2 - 7a-2z2 + 16a-2z4 - 13a-2z6 + 3a-2z8 - 2a-1z + 4a-1z3 + 5a-1z5 - 10a-1z7 + 3a-1z9 + 4 - 20z2 + 39z4 - 28z6 + 4z8 + z10 - az - 2az3 + 15az5 - 19az7 + 6az9 + 3a2 - 17a2z2 + 31a2z4 - 24a2z6 + 5a2z8 + a2z10 + a3z - a3z3 + a3z5 - 4a3z7 + 3a3z9 + a4 - 2a4z2 + 3a4z4 - 5a4z6 + 4a4z8 + 2a5z - 3a5z3 - 2a5z5 + 4a5z7 + a6z2 - 4a6z4 + 4a6z6 + a7z - 4a7z3 + 3a7z5 - a8z2 + a8z4

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

{0, 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 11174. 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 = 9                       1 j = 7                     2   j = 5                   3 1   j = 3                 5 2     j = 1               5 3       j = -1             7 5         j = -3           6 6           j = -5         5 6             j = -7       3 6               j = -9     2 5                 j = -11   1 3                   j = -13   2                     j = -15 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, 174]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 174]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[14, 5, 15, 6], X[16, 7, 17, 8], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[6, 15, 7, 16], > X[20, 17, 21, 18], X[8, 20, 9, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 174]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 174]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 2, 22, 6, 20, 8, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 174]][t]
Out[6]=	-4 5 11 15 2 3 4 -15 - t + -- - -- + -- + 15 t - 11 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 174]][z]
Out[7]=	4 6 8 1 - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 174]}
In[9]:=	{KnotDet[Knot[11, Alternating, 174]], KnotSignature[Knot[11, Alternating, 174]]}
Out[9]=	{79, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 174]][q]
Out[10]=	-7 3 5 8 11 12 12 2 3 4 -10 + q - -- + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 174]}
In[12]:=	A2Invariant[Knot[11, Alternating, 174]][q]
Out[12]=	-20 -18 -16 -14 -12 -10 2 3 -4 -2 2 4 1 + q - q + q - q - q + q - -- + -- - q + q - q + 2 q + 8 6 q q 8 12 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 174]][a, z]
Out[13]=	2 4 -2 2 4 2 3 z 2 2 4 2 4 z 4 - a - 3 a + a + 11 z - ---- - 12 a z + 4 a z + 9 z - -- - 2 2 a a 2 4 4 4 6 2 6 4 6 2 8 > 13 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 174]][a, z]
Out[14]=	2 -2 2 4 z 2 z 3 5 7 2 7 z 4 + a + 3 a + a - -- - --- - a z + a z + 2 a z + a z - 20 z - ---- - 3 a 2 a a 3 3 2 2 4 2 6 2 8 2 4 z 4 z 3 3 3 > 17 a z - 2 a z + a z - a z + ---- + ---- - 2 a z - a z - 3 a a 4 5 3 7 3 4 16 z 2 4 4 4 6 4 8 4 > 3 a z - 4 a z + 39 z + ----- + 31 a z + 3 a z - 4 a z + a z - 2 a 5 5 6 4 z 5 z 5 3 5 5 5 7 5 6 13 z > ---- + ---- + 15 a z + a z - 2 a z + 3 a z - 28 z - ----- - 3 a 2 a a 7 7 2 6 4 6 6 6 z 10 z 7 3 7 5 7 > 24 a z - 5 a z + 4 a z + -- - ----- - 19 a z - 4 a z + 4 a z + 3 a a 8 9 8 3 z 2 8 4 8 3 z 9 3 9 10 2 10 > 4 z + ---- + 5 a z + 4 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 174]], Vassiliev[3][Knot[11, Alternating, 174]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, Alternating, 174]][q, t]
Out[16]=	6 7 1 2 1 3 2 5 3 6 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 5 6 6 5 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a174

K11a173 K11a175