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

K11n16 K11n18

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X18,11,19,12 X20,13,21,14 X15,6,16,7 X22,17,1,18 X14,19,15,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 12t-1 - 19 + 12t - 2t2

Conway Polynomial:

1 + 4z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{47, -2}

Jones Polynomial:

- q-10 + 2q-9 - 4q-8 + 6q-7 - 7q-6 + 8q-5 - 7q-4 + 6q-3 - 4q-2 + 2q-1

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 - q-26 + 2q-22 - q-20 + q-18 + q-12 - q-10 + 2q-8 - q-4 + 2q-2

HOMFLY-PT Polynomial:

a2 + 2a2z2 - a4z4 - a6z4 + a8 + 2a8z2 - a10

Kauffman Polynomial:

- a2 + 3a2z2 - a3z + 3a3z3 + a3z5 + 2a4z2 - 3a4z4 + 3a4z6 - 3a5z + 6a5z3 - 8a5z5 + 4a5z7 - a6z4 - 5a6z6 + 3a6z8 - 3a7z + 12a7z3 - 16a7z5 + 3a7z7 + a7z9 + a8 - 4a8z2 + 14a8z4 - 17a8z6 + 5a8z8 - 5a9z + 17a9z3 - 12a9z5 + a9z9 + a10 - 5a10z2 + 12a10z4 - 9a10z6 + 2a10z8 - 4a11z + 8a11z3 - 5a11z5 + a11z7

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

{4, -10}

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 1117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   2 j = -3                 3 1 j = -5               3 1   j = -7             4 3     j = -9           4 3       j = -11         3 4         j = -13       3 4           j = -15     1 3             j = -17   1 3               j = -19   1                 j = -21 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, 17]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 17]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 16, 8, 17], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[15, 6, 16, 7], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 17]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -11, 7, -10, -8, 4, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 17]]
Out[5]=	DTCode[4, 8, 10, -16, 2, 18, 20, -6, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 17]][t]
Out[6]=	2 12 2 -19 - -- + -- + 12 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 17]][z]
Out[7]=	2 4 1 + 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 17]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 17]], KnotSignature[Knot[11, NonAlternating, 17]]}
Out[9]=	{47, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 17]][q]
Out[10]=	-10 2 4 6 7 8 7 6 4 2 -q + -- - -- + -- - -- + -- - -- + -- - -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 17]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 17]][q]
Out[12]=	-32 -30 -28 -26 2 -20 -18 -12 -10 2 -4 2 -q - q + q - q + --- - q + q + q - q + -- - q + -- 22 8 2 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 17]][a, z]
Out[13]=	2 8 10 2 2 8 2 4 4 6 4 a + a - a + 2 a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 17]][a, z]
Out[14]=	2 8 10 3 5 7 9 11 2 2 -a + a + a - a z - 3 a z - 3 a z - 5 a z - 4 a z + 3 a z + 4 2 8 2 10 2 3 3 5 3 7 3 9 3 > 2 a z - 4 a z - 5 a z + 3 a z + 6 a z + 12 a z + 17 a z + 11 3 4 4 6 4 8 4 10 4 3 5 5 5 > 8 a z - 3 a z - a z + 14 a z + 12 a z + a z - 8 a z - 7 5 9 5 11 5 4 6 6 6 8 6 10 6 > 16 a z - 12 a z - 5 a z + 3 a z - 5 a z - 17 a z - 9 a z + 5 7 7 7 11 7 6 8 8 8 10 8 7 9 9 9 > 4 a z + 3 a z + a z + 3 a z + 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 17]], Vassiliev[3][Knot[11, NonAlternating, 17]]}
Out[15]=	{4, -10}
In[16]:=	Kh[Knot[11, NonAlternating, 17]][q, t]
Out[16]=	-3 2 1 1 1 3 1 3 3 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q t q t q t q t q t q t q t 4 3 4 4 3 4 3 3 1 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 3 > ---- 3 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n17

K11n16 K11n18