9 19

9_18

9_20

Visit 9 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 19's page at Knotilus! Visit 9 19's page at the original Knot Atlas!

9 19 Quick Notes

9 19 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X11,18,12,1 X17,12,18,13
Gauss code	-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 10 14 2 18 16 6 12
Conway Notation	[23112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.0325 A-Polynomial See Data:9 19/A-polynomial

[edit Notes for 9 19's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-10t+17-10t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 0 }
Jones polynomial	$q^{4}-2q^{3}+4q^{2}-6q+7-7q^{-1}+6q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{4}-2z^{2}a^{-2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+3a^{2}z^{6}+2z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-7a^{3}z^{5}-11az^{5}-z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-11a^{2}z^{4}+z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+10az^{3}+z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+8a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}-a^{3}z-3az-za^{-1}+za^{-3}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+2q^{8}+q^{2}-1+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-3q^{70}-5q^{68}+16q^{66}-21q^{64}+24q^{62}-18q^{60}+2q^{58}+18q^{56}-34q^{54}+40q^{52}-31q^{50}+13q^{48}+11q^{46}-28q^{44}+33q^{42}-24q^{40}+8q^{38}+10q^{36}-23q^{34}+20q^{32}-6q^{30}-13q^{28}+30q^{26}-34q^{24}+27q^{22}-6q^{20}-20q^{18}+41q^{16}-51q^{14}+48q^{12}-25q^{10}-3q^{8}+30q^{6}-44q^{4}+44q^{2}-27+4q^{-2}+14q^{-4}-24q^{-6}+19q^{-8}-4q^{-10}-13q^{-12}+23q^{-14}-22q^{-16}+7q^{-18}+9q^{-20}-26q^{-22}+33q^{-24}-29q^{-26}+17q^{-28}-2q^{-30}-14q^{-32}+22q^{-34}-24q^{-36}+21q^{-38}-12q^{-40}+4q^{-42}+4q^{-44}-9q^{-46}+11q^{-48}-9q^{-50}+8q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_19.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+2q^{5}-q^{3}+q^{-1}-2q^{-3}+2q^{-5}-q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+6q^{26}-q^{24}-7q^{22}+7q^{20}+3q^{18}-10q^{16}+5q^{14}+6q^{12}-8q^{10}+q^{8}+5q^{6}-2q^{4}-4q^{2}+2+7q^{-2}-7q^{-4}-3q^{-6}+11q^{-8}-5q^{-10}-5q^{-12}+8q^{-14}-2q^{-16}-3q^{-18}+3q^{-20}-q^{-22}-q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+q^{53}+13q^{51}+2q^{49}-16q^{47}-10q^{45}+17q^{43}+20q^{41}-13q^{39}-30q^{37}+6q^{35}+33q^{33}+5q^{31}-35q^{29}-13q^{27}+34q^{25}+20q^{23}-27q^{21}-24q^{19}+21q^{17}+23q^{15}-13q^{13}-24q^{11}+5q^{9}+21q^{7}+6q^{5}-17q^{3}-17q+13q^{-1}+29q^{-3}-5q^{-5}-34q^{-7}-4q^{-9}+37q^{-11}+13q^{-13}-35q^{-15}-17q^{-17}+25q^{-19}+19q^{-21}-17q^{-23}-16q^{-25}+10q^{-27}+11q^{-29}-5q^{-31}-6q^{-33}+3q^{-35}+3q^{-37}-2q^{-39}-q^{-41}+2q^{-43}-q^{-47}-q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+6q^{94}-8q^{92}-13q^{90}-q^{88}+7q^{86}+31q^{84}+q^{82}-30q^{80}-28q^{78}-13q^{76}+57q^{74}+44q^{72}-7q^{70}-56q^{68}-81q^{66}+31q^{64}+85q^{62}+70q^{60}-25q^{58}-138q^{56}-53q^{54}+60q^{52}+142q^{50}+61q^{48}-130q^{46}-126q^{44}-14q^{42}+147q^{40}+128q^{38}-73q^{36}-141q^{34}-73q^{32}+105q^{30}+139q^{28}-17q^{26}-110q^{24}-87q^{22}+54q^{20}+112q^{18}+27q^{16}-69q^{14}-85q^{12}+q^{10}+74q^{8}+76q^{6}-15q^{4}-84q^{2}-75+20q^{-2}+131q^{-4}+61q^{-6}-60q^{-8}-145q^{-10}-61q^{-12}+138q^{-14}+132q^{-16}+10q^{-18}-155q^{-20}-132q^{-22}+79q^{-24}+134q^{-26}+76q^{-28}-88q^{-30}-131q^{-32}+6q^{-34}+69q^{-36}+82q^{-38}-16q^{-40}-73q^{-42}-17q^{-44}+9q^{-46}+43q^{-48}+9q^{-50}-23q^{-52}-6q^{-54}-9q^{-56}+12q^{-58}+5q^{-60}-4q^{-62}+4q^{-64}-6q^{-66}+q^{-68}-q^{-72}+4q^{-74}-q^{-76}-q^{-80}-q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+q^{143}+8q^{141}+13q^{139}+q^{137}-17q^{135}-22q^{133}-13q^{131}+13q^{129}+40q^{127}+44q^{125}-3q^{123}-57q^{121}-72q^{119}-39q^{117}+43q^{115}+114q^{113}+106q^{111}-123q^{107}-173q^{105}-104q^{103}+78q^{101}+233q^{99}+226q^{97}+30q^{95}-227q^{93}-348q^{91}-200q^{89}+147q^{87}+423q^{85}+394q^{83}+14q^{81}-429q^{79}-553q^{77}-225q^{75}+338q^{73}+664q^{71}+439q^{69}-191q^{67}-687q^{65}-607q^{63}+4q^{61}+633q^{59}+715q^{57}+168q^{55}-531q^{53}-746q^{51}-292q^{49}+397q^{47}+709q^{45}+376q^{43}-277q^{41}-635q^{39}-394q^{37}+172q^{35}+533q^{33}+392q^{31}-91q^{29}-442q^{27}-362q^{25}+22q^{23}+351q^{21}+348q^{19}+48q^{17}-267q^{15}-349q^{13}-139q^{11}+184q^{9}+362q^{7}+257q^{5}-75q^{3}-379q-403q^{-1}-65q^{-3}+381q^{-5}+543q^{-7}+243q^{-9}-324q^{-11}-667q^{-13}-442q^{-15}+217q^{-17}+733q^{-19}+619q^{-21}-50q^{-23}-705q^{-25}-752q^{-27}-143q^{-29}+598q^{-31}+799q^{-33}+301q^{-35}-412q^{-37}-740q^{-39}-423q^{-41}+218q^{-43}+610q^{-45}+453q^{-47}-48q^{-49}-430q^{-51}-413q^{-53}-72q^{-55}+261q^{-57}+325q^{-59}+124q^{-61}-130q^{-63}-221q^{-65}-125q^{-67}+41q^{-69}+133q^{-71}+103q^{-73}+2q^{-75}-71q^{-77}-68q^{-79}-17q^{-81}+30q^{-83}+40q^{-85}+20q^{-87}-9q^{-89}-23q^{-91}-14q^{-93}+q^{-95}+7q^{-97}+9q^{-99}+6q^{-101}-4q^{-103}-6q^{-105}-q^{-107}-2q^{-109}+q^{-111}+4q^{-113}+q^{-115}-q^{-117}-q^{-121}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+2q^{8}+q^{2}-1+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+40q^{36}-62q^{34}+86q^{32}-112q^{30}+129q^{28}-130q^{26}+120q^{24}-90q^{22}+48q^{20}+8q^{18}-70q^{16}+128q^{14}-177q^{12}+216q^{10}-234q^{8}+234q^{6}-214q^{4}+172q^{2}-122+64q^{-2}-9q^{-4}-38q^{-6}+78q^{-8}-94q^{-10}+104q^{-12}-100q^{-14}+92q^{-16}-78q^{-18}+62q^{-20}-50q^{-22}+36q^{-24}-26q^{-26}+17q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+2q^{32}-5q^{30}-q^{28}+5q^{26}+3q^{24}-5q^{22}-3q^{20}+6q^{18}+3q^{16}-5q^{14}-q^{12}+5q^{10}-q^{8}-2q^{6}+q^{4}-q^{2}-1+2q^{-2}+2q^{-4}-5q^{-6}-2q^{-8}+8q^{-10}+2q^{-12}-6q^{-14}+q^{-16}+7q^{-18}+q^{-20}-5q^{-22}-2q^{-24}+2q^{-26}-2q^{-30}+q^{-34}+q^{-36}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+2q^{28}-5q^{26}+4q^{24}+4q^{22}-6q^{20}+5q^{18}+5q^{16}-8q^{14}+q^{12}+4q^{10}-5q^{8}-q^{6}+3q^{4}+3q^{2}-1-q^{-2}+7q^{-4}-4q^{-6}-6q^{-8}+8q^{-10}-3q^{-12}-6q^{-14}+6q^{-16}-3q^{-20}+3q^{-22}+q^{-24}-q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+2q^{11}+q^{7}+q^{3}-q^{-1}+q^{-3}-2q^{-5}-q^{-9}+q^{-11}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+6q^{28}-7q^{26}+8q^{24}-8q^{22}+8q^{20}-5q^{18}+3q^{16}+2q^{14}-5q^{12}+10q^{10}-13q^{8}+15q^{6}-15q^{4}+15q^{2}-13+9q^{-2}-5q^{-4}+2q^{-8}-6q^{-10}+7q^{-12}-8q^{-14}+8q^{-16}-6q^{-18}+5q^{-20}-3q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+4q^{46}-2q^{44}-6q^{42}-q^{40}+8q^{38}+6q^{36}-5q^{34}-8q^{32}+2q^{30}+9q^{28}+4q^{26}-7q^{24}-6q^{22}+3q^{20}+6q^{18}-2q^{16}-6q^{14}+5q^{10}+q^{8}-5q^{6}-2q^{4}+6q^{2}+5-4q^{-2}-5q^{-4}+4q^{-6}+7q^{-8}-2q^{-10}-8q^{-12}-2q^{-14}+8q^{-16}+5q^{-18}-5q^{-20}-8q^{-22}+7q^{-26}+3q^{-28}-3q^{-30}-4q^{-32}+3q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-3q^{70}-5q^{68}+16q^{66}-21q^{64}+24q^{62}-18q^{60}+2q^{58}+18q^{56}-34q^{54}+40q^{52}-31q^{50}+13q^{48}+11q^{46}-28q^{44}+33q^{42}-24q^{40}+8q^{38}+10q^{36}-23q^{34}+20q^{32}-6q^{30}-13q^{28}+30q^{26}-34q^{24}+27q^{22}-6q^{20}-20q^{18}+41q^{16}-51q^{14}+48q^{12}-25q^{10}-3q^{8}+30q^{6}-44q^{4}+44q^{2}-27+4q^{-2}+14q^{-4}-24q^{-6}+19q^{-8}-4q^{-10}-13q^{-12}+23q^{-14}-22q^{-16}+7q^{-18}+9q^{-20}-26q^{-22}+33q^{-24}-29q^{-26}+17q^{-28}-2q^{-30}-14q^{-32}+22q^{-34}-24q^{-36}+21q^{-38}-12q^{-40}+4q^{-42}+4q^{-44}-9q^{-46}+11q^{-48}-9q^{-50}+8q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 19"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$2t^{2}-10t+17-10t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}-2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 41, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-2q^{3}+4q^{2}-6q+7-7q^{-1}+6q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{4}-2z^{2}a^{-2}-a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+3a^{2}z^{6}+2z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-7a^{3}z^{5}-11az^{5}-z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-11a^{2}z^{4}+z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+10az^{3}+z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+8a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}-a^{3}z-3az-za^{-1}+za^{-3}-a^{2}+a^{-2}+a^{-4}$

Vassiliev invariants

V2 and V3:

(-2, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $-8$ $32$ ${\frac {68}{3}}$ ${\frac {4}{3}}$ $64$ ${\frac {304}{3}}$ ${\frac {160}{3}}$ $-8$ $-{\frac {256}{3}}$ $32$ $-{\frac {544}{3}}$ $-{\frac {32}{3}}$ $-{\frac {991}{15}}$ $-{\frac {1676}{15}}$ ${\frac {5276}{45}}$ $-{\frac {113}{9}}$ ${\frac {449}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s+1$, where $s=$0 is the signature of 9 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	χ
9										1	1
7									1		-1
5								3	1		2
3							3	1			-2
1						4	3				1
-1					4	4					0
-3				2	3						-1
-5			2	4							2
-7		1	2								-1
-9		2									2
-11	1										-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 19]]
Out[2]=	9
In[3]:=	PD[Knot[9, 19]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[13, 16, 14, 17], X[7, 15, 8, 14], X[15, 7, 16, 6], X[11, 18, 12, 1], X[17, 12, 18, 13]]
In[4]:=	GaussCode[Knot[9, 19]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8]
In[5]:=	BR[Knot[9, 19]]
Out[5]=	BR[5, {1, -2, 1, -2, -2, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 19]][t]
Out[6]=	2 10 2 17 + -- - -- - 10 t + 2 t 2 t t
In[7]:=	Conway[Knot[9, 19]][z]
Out[7]=	2 4 1 - 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 19]}
In[9]:=	{KnotDet[Knot[9, 19]], KnotSignature[Knot[9, 19]]}
Out[9]=	{41, 0}
In[10]:=	J=Jones[Knot[9, 19]][q]
Out[10]=	-5 3 4 6 7 2 3 4 7 - q + -- - -- + -- - - - 6 q + 4 q - 2 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 19]}
In[12]:=	A2Invariant[Knot[9, 19]][q]
Out[12]=	-16 -14 -12 -10 2 -2 2 4 8 10 -1 - q + q + q - q + -- + q + q - 2 q + q - q + 8 q 12 14 q + q
In[13]:=	Kauffman[Knot[9, 19]][a, z]
Out[13]=	2 2 -4 -2 2 z z 3 2 2 z 3 z 2 2 a + a - a + -- - - - 3 a z - a z + 3 z - ---- - ---- + 8 a z + 3 a 4 2 a a a 3 3 4 4 2 3 z z 3 3 3 5 3 4 z 4 a z - ---- + -- + 10 a z + 4 a z - 2 a z - 4 z + -- - 3 a 4 a a 5 5 2 4 4 4 2 z z 5 3 5 5 5 6 11 a z - 8 a z + ---- - -- - 11 a z - 7 a z + a z + 2 z + 3 a a 6 7 2 z 2 6 4 6 2 z 7 3 7 8 2 8 ---- + 3 a z + 3 a z + ---- + 5 a z + 3 a z + z + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[9, 19]], Vassiliev[3][Knot[9, 19]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 19]][q, t]
Out[15]=	4 1 2 1 2 2 4 2 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 3 4 3 3 2 5 2 5 3 7 3 9 4 ---- + --- + 3 q t + 3 q t + q t + 3 q t + q t + q t + q t 3 q t q t