The Non Alternating Knot 10₁₆₃

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Acknowledgement

KnotPlot

Warning. In 1974 K. Perko noticed that the knots labeled 10₁₆₁ and 10₁₆₂ in Rolfsen's tables are in fact the same. In our table we removed his 10₁₆₂ and renumbered the subsequent knots, so that our 10 crossings total is 165, one less than Rolfsen's 166. Read more: 1 2 3 4.

PD Presentation: X₆₂₇₁ X₈₃₉₄ X_13,19,14,18 X_11,1,12,20 X_19,13,20,12 X_2,16,3,15 X_4,17,5,18 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9

Gauss Code: {1, -6, 2, -7, 8, -1, 9, -2, 10, -8, -4, 5, -3, -9, 6, -10, 7, 3, -5, 4}

DT (Dowker-Thistlethwaite) Code: 6 8 10 14 16 -20 -18 2 4 -12

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 1--2 3 3 / NotAvailable 1

Alexander Polynomial: t^-3 - 5t^-2 + 12t^-1 - 15 + 12t - 5t² + t³

Conway Polynomial: 1 + z² + z⁴ + z⁶

Other knots with the same Alexander/Conway Polynomial: {9₂₈, 9₂₉, K11n87, ...}

Determinant and Signature: {51, 2}

Jones Polynomial: - q^-2 + 4q^-1 - 6 + 8q - 9q² + 9q³ - 7q⁴ + 5q⁵ - 2q⁶

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

A2 (sl(3)) Invariant: - q^-6 + 2q^-4 + 1 + 2q² - 3q⁴ + q⁶ - 2q⁸ + 2q¹⁰ + 2q¹² + 2q¹⁶ - 2q¹⁸ - q²⁰

HOMFLY-PT Polynomial: - a^-6 + 2a^-4 - a^-4z⁴ - a^-2 + 2a^-2z² + 3a^-2z⁴ + a^-2z⁶ + 1 - z² - z⁴

Kauffman Polynomial: - 2a^-7z + 3a^-7z³ + a^-6 - 4a^-6z² + 4a^-6z⁴ + a^-6z⁶ - 3a^-5z + 7a^-5z³ - 4a^-5z⁵ + 3a^-5z⁷ + 2a^-4 - 4a^-4z² + a^-4z⁴ + 2a^-4z⁸ - a^-3z + 8a^-3z³ - 15a^-3z⁵ + 8a^-3z⁷ + a^-2 + 2a^-2z² - 11a^-2z⁴ + 3a^-2z⁶ + 2a^-2z⁸ + 3a^-1z³ - 10a^-1z⁵ + 5a^-1z⁷ + 1 + 2z² - 8z⁴ + 4z⁶ - az³ + az⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 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=2 is the signature of 10₁₆₃. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

j = 13 2

j = 11 3

j = 9 4 2

j = 7 5 3

j = 5 4 4

j = 3 4 5

j = 1 3 5

j = -1 1 3

j = -3 3

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-7 - 4q^-6 + q^-5 + 14q^-4 - 18q^-3 - 11q^-2 + 43q^-1 - 24 - 39q + 69q² - 15q³ - 66q⁴ + 78q⁵ - 78q⁷ + 67q⁸ + 13q⁹ - 66q¹⁰ + 39q¹¹ + 18q¹² - 35q¹³ + 11q¹⁴ + 10q¹⁵ - 8q¹⁶ + q¹⁸

3 - q^-15 + 4q^-14 - q^-13 - 9q^-12 - 4q^-11 + 21q^-10 + 23q^-9 - 40q^-8 - 51q^-7 + 38q^-6 + 106q^-5 - 23q^-4 - 160q^-3 - 26q^-2 + 212q^-1 + 94 - 236q - 185q² + 247q³ + 266q⁴ - 227q⁵ - 344q⁶ + 198q⁷ + 404q⁸ - 158q⁹ - 448q¹⁰ + 117q¹¹ + 468q¹² - 67q¹³ - 473q¹⁴ + 20q¹⁵ + 445q¹⁶ + 38q¹⁷ - 402q¹⁸ - 79q¹⁹ + 324q²⁰ + 115q²¹ - 238q²² - 124q²³ + 147q²⁴ + 114q²⁵ - 77q²⁶ - 80q²⁷ + 25q²⁸ + 47q²⁹ - 24q³¹ + 3q³³ + 4q³⁴ - 2q³⁵

4 q^-26 - 4q^-25 + q^-24 + 9q^-23 - q^-22 + q^-21 - 33q^-20 - 8q^-19 + 47q^-18 + 36q^-17 + 49q^-16 - 130q^-15 - 131q^-14 + 38q^-13 + 148q^-12 + 328q^-11 - 128q^-10 - 400q^-9 - 296q^-8 + 55q^-7 + 851q^-6 + 330q^-5 - 423q^-4 - 923q^-3 - 634q^-2 + 1109q^-1 + 1149 + 225q - 1298q² - 1745q³ + 703q⁴ + 1764q⁵ + 1316q⁶ - 1091q⁷ - 2693q⁸ - 126q⁹ + 1887q¹⁰ + 2306q¹¹ - 538q¹² - 3196q¹³ - 925q¹⁴ + 1691q¹⁵ + 2942q¹⁶ + 35q¹⁷ - 3322q¹⁸ - 1534q¹⁹ + 1350q²⁰ + 3243q²¹ + 574q²² - 3103q²³ - 1975q²⁴ + 806q²⁵ + 3155q²⁶ + 1131q²⁷ - 2414q²⁸ - 2147q²⁹ + 29q³⁰ + 2518q³¹ + 1511q³² - 1302q³³ - 1793q³⁴ - 650q³⁵ + 1404q³⁶ + 1364q³⁷ - 252q³⁸ - 975q³⁹ - 778q⁴⁰ + 386q⁴¹ + 746q⁴² + 193q⁴³ - 244q⁴⁴ - 420q⁴⁵ - 42q⁴⁶ + 199q⁴⁷ + 131q⁴⁸ + 17q⁴⁹ - 95q⁵⁰ - 45q⁵¹ + 13q⁵² + 20q⁵³ + 14q⁵⁴ - 5q⁵⁵ - 6q⁵⁶ + q⁵⁸

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]:=
PD[Knot[10, 163]]

Out[2]=
PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[13, 19, 14, 18], X[11, 1, 12, 20], > X[19, 13, 20, 12], X[2, 16, 3, 15], X[4, 17, 5, 18], X[10, 6, 11, 5], > X[14, 7, 15, 8], X[16, 10, 17, 9]]

In[3]:=
GaussCode[Knot[10, 163]]

Out[3]=
GaussCode[1, -6, 2, -7, 8, -1, 9, -2, 10, -8, -4, 5, -3, -9, 6, -10, 7, 3, -5, > 4]

In[4]:=
DTCode[Knot[10, 163]]

Out[4]=
DTCode[6, 8, 10, 14, 16, -20, -18, 2, 4, -12]

In[5]:=
br = BR[Knot[10, 163]]

Out[5]=
BR[4, {1, 1, -2, -1, -1, 3, 2, -1, 2, 2, 3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 163]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 163]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 163]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, {1, 2}, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 163]][t]

Out[10]=
-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t

In[11]:=
Conway[Knot[10, 163]][z]

Out[11]=
2 4 6 1 + z + z + z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}

In[13]:=
{KnotDet[Knot[10, 163]], KnotSignature[Knot[10, 163]]}

Out[13]=
{51, 2}

In[14]:=
Jones[Knot[10, 163]][q]

Out[14]=
-2 4 2 3 4 5 6 -6 - q + - + 8 q - 9 q + 9 q - 7 q + 5 q - 2 q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 163]}

In[16]:=
A2Invariant[Knot[10, 163]][q]

Out[16]=
-6 2 2 4 6 8 10 12 16 18 20 1 - q + -- + 2 q - 3 q + q - 2 q + 2 q + 2 q + 2 q - 2 q - q 4 q

In[17]:=
HOMFLYPT[Knot[10, 163]][a, z]

Out[17]=
2 4 4 6 -6 2 -2 2 2 z 4 z 3 z z 1 - a + -- - a - z + ---- - z - -- + ---- + -- 4 2 4 2 2 a a a a a

In[18]:=
Kauffman[Knot[10, 163]][a, z]

Out[18]=
2 2 2 3 3 -6 2 -2 2 z 3 z z 2 4 z 4 z 2 z 3 z 7 z 1 + a + -- + a - --- - --- - -- + 2 z - ---- - ---- + ---- + ---- + ---- + 4 7 5 3 6 4 2 7 5 a a a a a a a a a 3 3 4 4 4 5 5 5 8 z 3 z 3 4 4 z z 11 z 4 z 15 z 10 z > ---- + ---- - a z - 8 z + ---- + -- - ----- - ---- - ----- - ----- + 3 a 6 4 2 5 3 a a a a a a a 6 6 7 7 7 8 8 5 6 z 3 z 3 z 8 z 5 z 2 z 2 z > a z + 4 z + -- + ---- + ---- + ---- + ---- + ---- + ---- 6 2 5 3 a 4 2 a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 163]], Vassiliev[3][Knot[10, 163]]}

Out[19]=
{1, 2}

In[20]:=
Kh[Knot[10, 163]][q, t]

Out[20]=
3 1 3 1 3 3 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 4 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 13 5 > 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + 2 q t

In[21]:=
ColouredJones[Knot[10, 163], 2][q]

Out[21]=
-7 4 -5 14 18 11 43 2 3 4 -24 + q - -- + q + -- - -- - -- + -- - 39 q + 69 q - 15 q - 66 q + 6 4 3 2 q q q q q 5 7 8 9 10 11 12 13 > 78 q - 78 q + 67 q + 13 q - 66 q + 39 q + 18 q - 35 q + 14 15 16 18 > 11 q + 10 q - 8 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₆₃

10₁₆₂
10₁₆₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-7 - 4q^-6 + q^-5 + 14q^-4 - 18q^-3 - 11q^-2 + 43q^-1 - 24 - 39q + 69q² - 15q³ - 66q⁴ + 78q⁵ - 78q⁷ + 67q⁸ + 13q⁹ - 66q¹⁰ + 39q¹¹ + 18q¹² - 35q¹³ + 11q¹⁴ + 10q¹⁵ - 8q¹⁶ + q¹⁸
3	- q^-15 + 4q^-14 - q^-13 - 9q^-12 - 4q^-11 + 21q^-10 + 23q^-9 - 40q^-8 - 51q^-7 + 38q^-6 + 106q^-5 - 23q^-4 - 160q^-3 - 26q^-2 + 212q^-1 + 94 - 236q - 185q² + 247q³ + 266q⁴ - 227q⁵ - 344q⁶ + 198q⁷ + 404q⁸ - 158q⁹ - 448q¹⁰ + 117q¹¹ + 468q¹² - 67q¹³ - 473q¹⁴ + 20q¹⁵ + 445q¹⁶ + 38q¹⁷ - 402q¹⁸ - 79q¹⁹ + 324q²⁰ + 115q²¹ - 238q²² - 124q²³ + 147q²⁴ + 114q²⁵ - 77q²⁶ - 80q²⁷ + 25q²⁸ + 47q²⁹ - 24q³¹ + 3q³³ + 4q³⁴ - 2q³⁵
4	q^-26 - 4q^-25 + q^-24 + 9q^-23 - q^-22 + q^-21 - 33q^-20 - 8q^-19 + 47q^-18 + 36q^-17 + 49q^-16 - 130q^-15 - 131q^-14 + 38q^-13 + 148q^-12 + 328q^-11 - 128q^-10 - 400q^-9 - 296q^-8 + 55q^-7 + 851q^-6 + 330q^-5 - 423q^-4 - 923q^-3 - 634q^-2 + 1109q^-1 + 1149 + 225q - 1298q² - 1745q³ + 703q⁴ + 1764q⁵ + 1316q⁶ - 1091q⁷ - 2693q⁸ - 126q⁹ + 1887q¹⁰ + 2306q¹¹ - 538q¹² - 3196q¹³ - 925q¹⁴ + 1691q¹⁵ + 2942q¹⁶ + 35q¹⁷ - 3322q¹⁸ - 1534q¹⁹ + 1350q²⁰ + 3243q²¹ + 574q²² - 3103q²³ - 1975q²⁴ + 806q²⁵ + 3155q²⁶ + 1131q²⁷ - 2414q²⁸ - 2147q²⁹ + 29q³⁰ + 2518q³¹ + 1511q³² - 1302q³³ - 1793q³⁴ - 650q³⁵ + 1404q³⁶ + 1364q³⁷ - 252q³⁸ - 975q³⁹ - 778q⁴⁰ + 386q⁴¹ + 746q⁴² + 193q⁴³ - 244q⁴⁴ - 420q⁴⁵ - 42q⁴⁶ + 199q⁴⁷ + 131q⁴⁸ + 17q⁴⁹ - 95q⁵⁰ - 45q⁵¹ + 13q⁵² + 20q⁵³ + 14q⁵⁴ - 5q⁵⁵ - 6q⁵⁶ + q⁵⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 163]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[13, 19, 14, 18], X[11, 1, 12, 20], > X[19, 13, 20, 12], X[2, 16, 3, 15], X[4, 17, 5, 18], X[10, 6, 11, 5], > X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[3]:=	GaussCode[Knot[10, 163]]
Out[3]=	GaussCode[1, -6, 2, -7, 8, -1, 9, -2, 10, -8, -4, 5, -3, -9, 6, -10, 7, 3, -5, > 4]
In[4]:=	DTCode[Knot[10, 163]]
Out[4]=	DTCode[6, 8, 10, 14, 16, -20, -18, 2, 4, -12]
In[5]:=	br = BR[Knot[10, 163]]
Out[5]=	BR[4, {1, 1, -2, -1, -1, 3, 2, -1, 2, 2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 163]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 163]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 163]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {1, 2}, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 163]][t]
Out[10]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[10, 163]][z]
Out[11]=	2 4 6 1 + z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[13]:=	{KnotDet[Knot[10, 163]], KnotSignature[Knot[10, 163]]}
Out[13]=	{51, 2}
In[14]:=	Jones[Knot[10, 163]][q]
Out[14]=	-2 4 2 3 4 5 6 -6 - q + - + 8 q - 9 q + 9 q - 7 q + 5 q - 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 163]}
In[16]:=	A2Invariant[Knot[10, 163]][q]
Out[16]=	-6 2 2 4 6 8 10 12 16 18 20 1 - q + -- + 2 q - 3 q + q - 2 q + 2 q + 2 q + 2 q - 2 q - q 4 q
In[17]:=	HOMFLYPT[Knot[10, 163]][a, z]
Out[17]=	2 4 4 6 -6 2 -2 2 2 z 4 z 3 z z 1 - a + -- - a - z + ---- - z - -- + ---- + -- 4 2 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[10, 163]][a, z]
Out[18]=	2 2 2 3 3 -6 2 -2 2 z 3 z z 2 4 z 4 z 2 z 3 z 7 z 1 + a + -- + a - --- - --- - -- + 2 z - ---- - ---- + ---- + ---- + ---- + 4 7 5 3 6 4 2 7 5 a a a a a a a a a 3 3 4 4 4 5 5 5 8 z 3 z 3 4 4 z z 11 z 4 z 15 z 10 z > ---- + ---- - a z - 8 z + ---- + -- - ----- - ---- - ----- - ----- + 3 a 6 4 2 5 3 a a a a a a a 6 6 7 7 7 8 8 5 6 z 3 z 3 z 8 z 5 z 2 z 2 z > a z + 4 z + -- + ---- + ---- + ---- + ---- + ---- + ---- 6 2 5 3 a 4 2 a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 163]], Vassiliev[3][Knot[10, 163]]}
Out[19]=	{1, 2}
In[20]:=	Kh[Knot[10, 163]][q, t]
Out[20]=	3 1 3 1 3 3 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 4 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 13 5 > 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + 2 q t
In[21]:=	ColouredJones[Knot[10, 163], 2][q]
Out[21]=	-7 4 -5 14 18 11 43 2 3 4 -24 + q - -- + q + -- - -- - -- + -- - 39 q + 69 q - 15 q - 66 q + 6 4 3 2 q q q q q 5 7 8 9 10 11 12 13 > 78 q - 78 q + 67 q + 13 q - 66 q + 39 q + 18 q - 35 q + 14 15 16 18 > 11 q + 10 q - 8 q + q

The Non Alternating Knot 10163

The Non Alternating Knot 10₁₆₃